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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Fundamental Theorem of Calculus</h3>
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<h2 class="hd hd-2 unit-title">2. The first fundamental theorem of calculus</h2>
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<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Know the statement of the <span style="color:#99182C"><b class="bf">first fundamental theorem of calculus</b></span> and use it to evaluate definite integrals. </p></li><li><p>
Use <span style="color:#99182C"><b class="bf">properties of definite integrals</b></span>, such as additivity and symmetry, to set up and evaluate definite integrals. </p></li><li><p>
Use the <span style="color:#99182C"><b class="bf">method of substitution</b></span> to evaluate definite integrals. </p></li></ul><p><b class="bfseries">Contents: 22 pages</b></p><p>
11 videos (59 minutes 1x speed) 39 questions </p>
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<h2 class="hd hd-2 unit-title">3. Exploration</h2>
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Integral of the constant revisited
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Let [mathjaxinline]F(x)= mx[/mathjaxinline] for some constant [mathjaxinline]m&gt;0[/mathjaxinline]. And let [mathjaxinline]f(x)= F'(x)=m[/mathjaxinline]. <br/></p>
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Which of the following equals [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]? <br/></p>
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(Check all that apply.)<br/></p>
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<text>Area under [mathjaxinline]y=f(x)[/mathjaxinline] between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]</text>
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<text>Area under [mathjaxinline]y=F(x)[/mathjaxinline] between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]</text>
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<text>[mathjaxinline]m[/mathjaxinline]</text>
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<text>[mathjaxinline]b-a[/mathjaxinline]</text>
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<text>[mathjaxinline]m\cdot (b-a)[/mathjaxinline]</text>
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<text>[mathjaxinline]F(a)-F(b)[/mathjaxinline]</text>
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<text>[mathjaxinline]F(b)-F(a)[/mathjaxinline]</text>
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<text>[mathjaxinline]f(b)-f(a)[/mathjaxinline]</text>
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Constant of integration
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Given a function [mathjaxinline]f[/mathjaxinline]. Let [mathjaxinline]F[/mathjaxinline] and [mathjaxinline]G[/mathjaxinline] be two different antiderivatives of [mathjaxinline]f[/mathjaxinline], That is, [mathjaxinline]F'=G'=f[/mathjaxinline] but [mathjaxinline]F\neq G[/mathjaxinline]. Which of the following are true?<br/>(Check all that apply.)<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory2-tab3-problem2_2_1">
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<input type="checkbox" name="input_theory2-tab3-problem2_2_1[]" id="input_theory2-tab3-problem2_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="theory2-tab3-problem2_2_1-choice_1-label" for="input_theory2-tab3-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory2-tab3-problem2_2_1"> <text>[mathjaxinline]F(b)-F(a)=G(b)-G(a)[/mathjaxinline]</text>
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<input type="checkbox" name="input_theory2-tab3-problem2_2_1[]" id="input_theory2-tab3-problem2_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="theory2-tab3-problem2_2_1-choice_2-label" for="input_theory2-tab3-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory2-tab3-problem2_2_1"> <text>[mathjaxinline]F(b)-G(b)=F(a)-G(a)=C\,[/mathjaxinline] for some constant [mathjaxinline]\, C\neq 0[/mathjaxinline]</text>
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<h3 class="hd hd-2">The First Fundamental Theorem of Calculus</h3>
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<p><b class="bfseries">The First Fundamental Theorem of Calculus</b></p><p>
The <span style="color:#99182C"><b class="bf">First Fundamental Theorem of Calculus</b></span> states that:<br/>If [mathjaxinline]F[/mathjaxinline] is differentiable, and [mathjaxinline]F'=f[/mathjaxinline] is continuous, then </p><table id="a0000000889" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000890"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F(b)-F(a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.89)</td></tr></table><p>
In other words, the definite integral of a function is the difference between the values of its antiderivative at the limits of the definite integral.<br/></p><p>
We will abbreviate the <span style="color:#99182C"><b class="bf">First Fundamental Theorem of Calculus</b></span> as <span style="color:#99182C"><b class="bf">FTC1</b></span>.<br/></p><p>
The FTC1 connects the definite integral to the antiderivative. With this connection, we can now compute definite integrals using antiderivatives, and dispense with Riemann sums. </p><p>
Notice that in the first problem, you have checked that this statement is true when [mathjaxinline]f(x)[/mathjaxinline] is equal to a constant, [mathjaxinline]m[/mathjaxinline].<br/></p>
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<p><b class="bfseries">Notation</b></p><p>
Here is a new notation for denoting the difference of the values of a function at two points. </p><table id="a0000000891" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000892"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \left.\phantom{\int }F(x)\, \right|_ a^ b[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle F(b)-F(a)[/mathjaxinline]
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Notice the order of the difference. This is the function evaluated at [mathjaxinline]b[/mathjaxinline], written at the top of the vertical bar, minus the function evaluated at [mathjaxinline]a[/mathjaxinline], at the bottom of the vertical bar. </p>
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FTC in new notation
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In this new notation, the first fundamental theorem of calculus is </p>
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<text> [mathjaxinline]\displaystyle \left.\int _ a^ b F(x)\, dx = F(x)\, \right|_ b^ a[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \left.\int _ a^ b F(x)\, dx = F(x)\, \right|_ a^ b[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \left.\int _ a^ b F'(x)\, dx = F(x)\, \right|_ a^ b[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \left.\int _ a^ b F'(x)\, dx = F'(x)\, \right|_ a^ b[/mathjaxinline]</text>
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In the last section, we evaluated [mathjaxinline]\displaystyle \int _{0}^{1} e^ x dx[/mathjaxinline] as the limit of a Riemann sum.<br/></p>
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<p style="display:inline">Now, by FTC1, for [mathjaxinline]b\geq 0[/mathjaxinline], [mathjaxinline]\displaystyle \int _{0}^{b} e^ x dx=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
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<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<p style="display:inline">By FTC1, [mathjaxinline]\displaystyle \int _{1}^{2} w^4 dw=[/mathjaxinline]</p>
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Practice 3
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<p style="display:inline">By FTC1, [mathjaxinline]\displaystyle \int _{-\pi /2}^{\pi /2} \cos (\theta ) \, d\theta =[/mathjaxinline]</p>
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Is it true that [mathjaxinline]\displaystyle \int _{-\pi /2}^{\pi /2} \cos (\theta )\, d\theta = \int _{0}^{\pi } \sin (\theta ) \, d\theta[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory2-tab4-problem3_3_1">
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<h2 class="hd hd-2 unit-title">5. More practice</h2>
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<h3 class="hd hd-3 problem-header" id="theory2-tab5-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory2-tab5-problem1-problem-progress" tabindex="-1">
Practice 4
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<p style="display:inline">By FTC1, [mathjaxinline]\displaystyle \int ^{\frac{1}{2}}_{0} \frac{2}{\sqrt {1-y^2}} \, dy=[/mathjaxinline]</p>
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Practice 5
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<p style="display:inline">By FTC1, [mathjaxinline]\displaystyle \int _{1}^{2} \left(\frac{1}{2+2t^2} +\frac{3}{t}\right)\, dt=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory2-tab5-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<div class="formulainput">
<table class="formulainput">
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
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Practice 6
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<h2 class="hd hd-2 unit-title">6. The intuition: velocity versus speed</h2>
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In the following video, we have used the words “velocity" and “speed" interchangeably because we have assumed that we were always traveling forwards.<br/></p><p>
Recall that if the velocity function is [mathjaxinline]v(t)[/mathjaxinline], then the <span style="color:#99182C"><b class="bf">speed</b></span> function is [mathjaxinline]|v(t)|[/mathjaxinline]. In other words, speed is the absolute value of velocity.<br/></p><p>
In the case when [mathjaxinline]v(t)>0[/mathjaxinline] for all [mathjaxinline]t[/mathjaxinline], [mathjaxinline]|v(t)|=v(t)[/mathjaxinline], so the speed and velocity are the same. </p>
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<p><b class="bfseries">Intuition: traveling in one direction</b></p><p>
Suppose you are traveling always in one direction between time [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline], and the velocity of your car at time [mathjaxinline]t[/mathjaxinline] is [mathjaxinline]v(t)\geq 0[/mathjaxinline], and the position of your car at time [mathjaxinline]t[/mathjaxinline] is [mathjaxinline]x(t)[/mathjaxinline].<br/></p><p>
If you record your velocity every second. In other words, let [mathjaxinline]t_ i[/mathjaxinline] be the moment within the [mathjaxinline]i^{\text {th}}[/mathjaxinline] second when you read the speedometer, and [mathjaxinline]v(t_ i)[/mathjaxinline] be the velocity of your car at [mathjaxinline]t_ i[/mathjaxinline]. And let [mathjaxinline]\Delta t[/mathjaxinline] be one second. Then [mathjaxinline]v(t_ i)\cdot \Delta t[/mathjaxinline] is an approximation of the distance travelled within the [mathjaxinline]i^{\text {th}}[/mathjaxinline] second, and when we add up all these small distances, we get an approximation of the total distance travelled in the entire journey.<br/></p><table id="a0000000918" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000919"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sum _{i=1}^{n} v(t_ i) \cdot \Delta t[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(b)-x(a) \qquad (\text {Riemann sum approximation}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.106)</td></tr></table><p>
In other words, the Riemann sum <b class="bfseries">approximates</b> the total distance travelled by your car in the journey.</p><p>
On the other hand, the first fundamental theorem says </p><table id="a0000000920" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000921"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b v(t)\, dt[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(b)-x(a) \qquad (\text {FTC1}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.107)</td></tr></table><p>
That is, the definite integral is <b class="bfseries">equal</b> to the total distance travelled in the whole journey.</p>
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<h2 class="hd hd-2 unit-title">7. Preparation for definite integrals of general functions</h2>
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<p>
So far, we have only defined and discussed definite integrals for [mathjaxinline]f\geq 0[/mathjaxinline]. Geometrically, this is the area under the graph of [mathjaxinline]f[/mathjaxinline].</p><p>
But what about definite integrals of general functions, which can be negative? <br/></p><p>
Recall the statement of FTC1: </p><table id="a0000000922" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000923"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle F(b)-F(a)\qquad \text {where}\, \, F'(x)=f(x),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.108)</td></tr></table><p>
Notice that the right hand side of the above equation, [mathjaxinline]F(b)-F(a)[/mathjaxinline], is still defined even if [mathjaxinline]f(x)[/mathjaxinline] is sometimes negative. This is because [mathjaxinline]F(x)[/mathjaxinline] is an antiderivative of [mathjaxinline]f[/mathjaxinline], and antiderivatives are defined for general functions, not only the non-negative ones.<br/></p><p>
Therefore, we can use this equation to define [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline] for a general function [mathjaxinline]f[/mathjaxinline]. That is, for <b class="bfseries">any</b> continuous function [mathjaxinline]f[/mathjaxinline], let </p><table id="a0000000924" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000925"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle F(b)-F(a),\qquad \text {where}\, \, F'(x)=f(x).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.109)</td></tr></table><p>
In following problems, we will apply the equation in FTC1 to evaluate the definite integral of functions that can be negative somewhere.</p>
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Integral of a negative constant function
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<p style="display:inline">Let [mathjaxinline]f(t)=-5[/mathjaxinline]. Then by FTC1, [mathjaxinline]\displaystyle \int _{4}^{8} f(t)\, dt=[/mathjaxinline]</p>
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Interpret the integral of a negative constant function
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Let us think of what the definite integral in previous problems mean in terms of area.<br/></p>
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Which of the following equals [mathjaxinline]\displaystyle \int _{4}^{8} f(t) dt[/mathjaxinline]?<br/></p>
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<text> Area of the rectangle below [mathjaxinline]y=-5[/mathjaxinline], between [mathjaxinline]t=4[/mathjaxinline] and [mathjaxinline]t=8[/mathjaxinline]</text>
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<text> Area of the rectangle below the [mathjaxinline]t[/mathjaxinline]-axis, above [mathjaxinline]y=-5[/mathjaxinline], between [mathjaxinline]t=4[/mathjaxinline] and [mathjaxinline]t=8[/mathjaxinline]</text>
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<text> [mathjaxinline]-\, (\,[/mathjaxinline]Area of rectangle BELOW the [mathjaxinline]t[/mathjaxinline]-axis, above [mathjaxinline]y=-5[/mathjaxinline], between [mathjaxinline]t=4[/mathjaxinline] and [mathjaxinline]t=8\, )[/mathjaxinline]</text>
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Interpret the integral of a negative constant velocity function
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Suppose that [mathjaxinline]f(t)=-5\, \text {km}/\text {hour}[/mathjaxinline] represents the velocity of your car. In other words, you are travelling in the negative direction with constant speed. <br/></p>
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What does [mathjaxinline]\displaystyle \int _{4}^{8} f(t) \, dt[/mathjaxinline] represent?<br/>(Check all that apply.)<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory2-tab7-problem3_2_1">
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<input type="checkbox" name="input_theory2-tab7-problem3_2_1[]" id="input_theory2-tab7-problem3_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory2-tab7-problem3_2_1-choice_3-label" for="input_theory2-tab7-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory2-tab7-problem3_2_1"> <text>Direction and total distance travelled from [mathjaxinline]t=4[/mathjaxinline] to [mathjaxinline]t=8[/mathjaxinline]</text>
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Traveling roundtrip
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Suppose you drove along a straight road from home to the border, which is 125 km away. You took 2 hours to drive there but just as you arrived you realized that you had forgotten to bring your passport, and so you immediately turned back to drive home again. The round trip took 4 hours in total. </p>
<p>
Let [mathjaxinline]v(t)[/mathjaxinline] be your car's velocity (in km/hour) and let [mathjaxinline]x(t)[/mathjaxinline] be your car's position (in km) at time [mathjaxinline]t[/mathjaxinline] (in hours). Their graphs are given below. </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:center; border:none">
<img alt="The graph of v of t versus t is plotted. The function starts at the origin and increases rapidly to a height of 65 where it plateaus and remains constant until approximately t equals 2 when the function decreasing sharply to a height of negative 65. The function remains constant again until approximately t equals 4 when the function increasing sharply to a height of 0." src="/assets/courseware/v1/aa17818d90e224259caf285eb7ca7d16/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_roundtrip.svg" style="margin: 10px 25px 25px 25px" width="280px"/>
</td>
<td style="text-align:center; border:none">&#160;</td>
<td style="text-align:center; border:none">
<img alt="The graph of x of t versus t is plotted. The function starts at the origin and increases to a maximum of 125 at t equals 2 then decreases to 0 at t equals 4." src="/assets/courseware/v1/a58476ecf4576df86717cfc490657b92/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_roundtripx.svg" style="margin: 10px 25px 25px 25px" width="280px"/>
</td>
</tr>
<tr>
<td style="text-align:center; border:none">
time-velocity graph </td>
<td style="text-align:center; border:none">&#160;</td>
<td style="text-align:center; border:none">
time-position graph</td>
</tr>
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<p>
Applying FTC1, <p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^{2} v(t)\, dt=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory2-tab7-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Therefore, [mathjaxinline]\displaystyle \int _{0}^{4} v(t)\, dt[/mathjaxinline] can be interpreted as<br/></p>
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<text> Total distance the car has travelled between time [mathjaxinline]0[/mathjaxinline] and [mathjaxinline]4[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">8. Definite integrals for general functions</h2>
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<p><b class="bfseries">The definite integral of general functions</b></p><p>
Since the equation in the statement of FTC1 makes sense for general functions, we can use the FTC1 to extend the definition of the definite integral to functions that are not necessarily non-negative. That is:<br/>For any continuous function [mathjaxinline]f[/mathjaxinline] with an antiderivative [mathjaxinline]F[/mathjaxinline], </p><table id="a0000000935" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000936"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle F(b)-F(a)\qquad \left(\text {for any continuous}\, \, f(x)=F'(x)\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.116)</td></tr></table><p>
It turns out that to be consistent with FTC1, the Riemann sum formula for definite integrals does not need to change. In other words, the definite integral of any continuous function [mathjaxinline]f[/mathjaxinline] (not necessarily non-negative), from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]b[/mathjaxinline], is </p><table id="a0000000937" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x) \, dx\, =\, \lim _{n\rightarrow \infty }\, \sum _{i=1}^{n} \, f(c_ i) \Delta x\qquad (\text {for any continuous}\, \, \, f(x)).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Here, as before, [mathjaxinline]\displaystyle \Delta x= \frac{b-a}{n}[/mathjaxinline] is the length of any one of the [mathjaxinline]n[/mathjaxinline] subintervals that [mathjaxinline][a,b][/mathjaxinline] is divided into, and [mathjaxinline]c_ i[/mathjaxinline] is any point within the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval.<br/></p><center><img src="/assets/courseware/v1/d59a11e217e2a8f9dd8bfa2bb3895d16/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_Rimannnegative.svg" width="300px" alt=" " style="margin: 10px 25px 25px 25px"/><br/></center><p>
The only difference from the case when [mathjaxinline]f\geq 0[/mathjaxinline] is that [mathjaxinline]f(c_ i)[/mathjaxinline] can now also be negative. Therefore, [mathjaxinline]f(c_ i)[/mathjaxinline] is not just the height of the [mathjaxinline]i^{\text {th}}[/mathjaxinline] rectangle, but the height with a sign.<br/></p><p>
Consequently, the geometric definition of [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline] that is consistent with FTC1 is as follows. </p><center><img src="/assets/courseware/v1/2ebc4ef59dc3ea3cc671f1d22d6de4f3/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_signedarea.svg" width="300px" alt=" " style="margin: 10px 25px 25px 25px"/><br/></center><table id="a0000000938" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x) \, dx\, =\, \text {Area above} \, x\text {-axis and below }\, y=f(x)\, -\, \text {Area below} \, x\text {-axis and above }\, y=f(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where the areas considered are between the vertical lines [mathjaxinline]x=a[/mathjaxinline] and [mathjaxinline]x=b[/mathjaxinline]. The important point is that the area above the [mathjaxinline]x[/mathjaxinline]-axis is counted with a positive sign and the area below the [mathjaxinline]x[/mathjaxinline]-axis is counted with a negative sign. In other words, the definite integral of a general function is the <span style="color:#99182C"><b class="bf">signed area bounded by the curve</b></span> <span style="color:#99182C"><p>
[mathjaxinline]y=f(x)[/mathjaxinline].<br/></p><p>
We will continue to use FTC1, not Riemann sums, to evaluate definite integrals. We will also often use the area interpretation to deduce properties of definite integrals, for example when looking for symmetry. </p></span></p>
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Traveling roundtrip again
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Consider the same roundtrip as before. That is, you drove 125 km on a straight road from your home to the border, but had to turn around when you arrive because you have forgotten to bring your passport. You drove 2 hours on the same road back home. <br/></p>
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As before, let [mathjaxinline]v(t)[/mathjaxinline] be your car's velocity (in km/hour) at time [mathjaxinline]t[/mathjaxinline] (in hours), and let [mathjaxinline]x(t)[/mathjaxinline] be your car's position (in km) at time [mathjaxinline]t[/mathjaxinline]. Their graphs are the same as before.<br/></p>
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<img alt="The graph of v of t versus t is plotted. The function starts at the origin and increases rapidly to a height of 65 where it plateaus and remains constant until approximately t equals 2 when the function decreasing sharply to a height of negative 65. The function remains constant again until approximately t equals 4 when the function increasing sharply to a height of 0. The region above the horizontal axis, below v of t, and between t equals 0 and t equals 2 is shaded blue and labeled A. The region below the horizontal axis, above v of t, and between t equals 2 and t equals 4 is shaded pink and labeled B." src="/assets/courseware/v1/82eff49c4c7b1ae52778993e655d6f1d/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_roundtriparea.svg" style="margin: 10px 25px 25px 25px" width="280px"/>
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<img alt="The graph of x of t versus t is plotted. The function starts at the origin and increases to a maximum of 125 at t equals 2 then decreases to 0 at t equals 4." src="/assets/courseware/v1/a58476ecf4576df86717cfc490657b92/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_roundtripx.svg" style="margin: 10px 25px 25px 25px" width="280px"/>
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time-velocity graph</td>
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time-position graph</td>
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Let [mathjaxinline]A&gt;0[/mathjaxinline] be the area of the blue region, and [mathjaxinline]B&gt;0[/mathjaxinline] be the area of the pink region. (These are geometric areas, hence positive numbers, not signed areas.)<br/>Using FTC1, give the geometric interpretation of the following in terms of the areas.<br/>(Enter your answers in terms of [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline]. Do NOT enter numerical answers)<br/></p>
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<p style="display:inline">[mathjaxinline]\displaystyle x(2)-x(0)=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle x(4)-x(2)=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle x(4)-x(0)=[/mathjaxinline]</p>
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Interpret the integral of speed
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The <span style="color:#99182C"><b class="bf">speed</b></span> is the absolute value of the velocity, [mathjaxinline]|v(t)|[/mathjaxinline].<br/></p>
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Consider the same roundtrip as before. That is, you drove 125 km on a straight road from your home to the border, but had to turn around when you arrive because you have forgotten to bring your passport. You drove 2 hours on the same road back home. <br/></p>
<p>
As above, let [mathjaxinline]v(t)[/mathjaxinline] be your car's velocity (in km/hour) at time [mathjaxinline]t[/mathjaxinline] (in hours), and let [mathjaxinline]x(t)[/mathjaxinline] be your car's position (in km) at time [mathjaxinline]t[/mathjaxinline]. Their graphs are the same as before.<br/></p>
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<img alt="The graph of v of t versus t is plotted. The function starts at the origin and increases rapidly to a height of 65 where it plateaus and remains constant until approximately t equals 2 when the function decreasing sharply to a height of negative 65. The function remains constant again until approximately t equals 4 when the function increasing sharply to a height of 0. The region above the horizontal axis, below v of t, and between t equals 0 and t equals 2 is shaded blue and labeled A. The region below the horizontal axis, above v of t, and between t equals 2 and t equals 4 is shaded pink and labeled B." src="/assets/courseware/v1/82eff49c4c7b1ae52778993e655d6f1d/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_roundtriparea.svg" style="margin: 10px 25px 25px 25px" width="280px"/>
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<img alt="The graph of x of t versus t is plotted. The function starts at the origin and increases to a maximum of 125 at t equals 2 then decreases to 0 at t equals 4." src="/assets/courseware/v1/a58476ecf4576df86717cfc490657b92/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_roundtripx.svg" style="margin: 10px 25px 25px 25px" width="280px"/>
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time-velocity graph</td>
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time-position graph</td>
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As above, let [mathjaxinline]A&gt;0[/mathjaxinline] be the area of the blue region, and [mathjaxinline]B&gt;0[/mathjaxinline] be the area of the pink region. (These are geometric areas, hence positive numbers, not signed areas.)<br/></p>
<p>
Which of the following equals [mathjaxinline]\displaystyle \int _{0}^{4} |v(t)|\, dt=?[/mathjaxinline]<br/>(Check all that apply.)<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory2-tab9-problem2_2_1">
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<input type="checkbox" name="input_theory2-tab9-problem2_2_1[]" id="input_theory2-tab9-problem2_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory2-tab9-problem2_2_1-choice_3-label" for="input_theory2-tab9-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory2-tab9-problem2_2_1"> <text>[mathjaxinline]\displaystyle \int _{0}^{2} v(t) \, dt+\int _{2}^{4} v(t)\, dt[/mathjaxinline]</text>
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<input type="checkbox" name="input_theory2-tab9-problem2_2_1[]" id="input_theory2-tab9-problem2_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="theory2-tab9-problem2_2_1-choice_4-label" for="input_theory2-tab9-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory2-tab9-problem2_2_1"> <text>[mathjaxinline]\displaystyle \int _{0}^{2} v(t) \, dt-\int _{2}^{4} v(t)\, dt[/mathjaxinline]</text>
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Evaluate the integral of speed
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Consider the same roundtrip as before. That is, you drove 125 km on a straight road from your home to the border, but had to turn around when you arrive because you have forgotten to bring your passport. You drove 2 hours on the same road back home. <br/></p>
<p>
As before, let [mathjaxinline]v(t)[/mathjaxinline] be your car's velocity (in km/hour) at time [mathjaxinline]t[/mathjaxinline] (in hours), and let [mathjaxinline]x(t)[/mathjaxinline] be your car's position (in km) at time [mathjaxinline]t[/mathjaxinline]. Their graphs are the same as before.<br/></p>
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<img alt="The graph of v of t versus t is plotted. The function starts at the origin and increases rapidly to a height of 65 where it plateaus and remains constant until approximately t equals 2 when the function decreasing sharply to a height of negative 65. The function remains constant again until approximately t equals 4 when the function increasing sharply to a height of 0. The region above the horizontal axis, below v of t, and between t equals 0 and t equals 2 is shaded blue. The region below the horizontal axis, above v of t, and between t equals 2 and t equals 4 is shaded pink ." src="/assets/courseware/v1/2eb2c6ae3bc038727d01db9263790407/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_roundtripareanoAB.svg" style="margin: 10px 25px 25px 25px" width="250px"/>
</td>
<td style="text-align:center; border:none">&#160;</td>
<td style="text-align:center; border:none">
<img alt="The graph of x of t versus t is plotted. The function starts at the origin and increases to a maximum of 125 at t equals 2 then decreases to 0 at t equals 4." src="/assets/courseware/v1/a58476ecf4576df86717cfc490657b92/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_roundtripx.svg" style="margin: 10px 25px 25px 25px" width="250px"/>
</td>
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<tr>
<td style="text-align:center; border:none">
time-velocity graph </td>
<td style="text-align:center; border:none">&#160;</td>
<td style="text-align:center; border:none">
time-position graph</td>
</tr>
</table>
<p>
Which of the following equals [mathjaxinline]\displaystyle \int _{0}^{4} |v(t)|\, dt=?[/mathjaxinline]<br/>(Check all that apply. You must choose at least one answer from each column.)<br/></p>
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<text>[mathjaxinline]x(4)-x(0)[/mathjaxinline]</text>
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For the problems below, you may enter your numerical answers as formulas evaluated at different values. For example, enter arctan(2) for the number [mathjaxinline]\arctan (2)[/mathjaxinline]. You may also enter numerical values, especially for formulas which are easy to evaluate. </p>
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Practice 1
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Given the graph of [mathjaxinline]f(x)[/mathjaxinline] below. </p>
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<img alt="A piecewise linear function y of x is plotted. The function starts at the point negative 1 comma 1. It decreases linearly to the origin. The function is then constant until the point 0 comma 1. The function then decreases linearly to the point 2 comma negative 1. The function is then constant until the point 3 comma negative 1. The function then decreases linearly to the point 4 comma negative 2." src="/assets/courseware/v1/f55212cf3d3af897d6a5da53ed748cd0/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_staircasedown.svg" style="margin: 10px 25px 25px 25px" width="350px"/>
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<p style="display:inline"> [mathjaxinline]\displaystyle \int _{1}^{4} f(x) dx=[/mathjaxinline]</p>
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<p style="display:inline"> [mathjaxinline]\displaystyle \int _{-1}^{4} f(x) dx=[/mathjaxinline]</p>
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Practice 2
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-1}^{10} x\, dx=[/mathjaxinline]</p>
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Practice 3
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-977}^{977} x^5 \, dx=[/mathjaxinline]</p>
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Odd functions
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Suppose [mathjaxinline]g(t)[/mathjaxinline] is an odd function. Recall that [mathjaxinline]g(t)[/mathjaxinline] is odd if [mathjaxinline]g(-t)=-g(t)[/mathjaxinline].<br/></p>
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Practice 4
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-\pi /2}^{39\pi /2} \cos (\theta ) \, d\theta =[/mathjaxinline]</p>
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<p style="display:inline">By FTC1, [mathjaxinline]\displaystyle \int _{-e^2}^{-e} \frac{-1}{t} dt=[/mathjaxinline]</p>
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<p><b class="bfseries">Properties of definite integrals</b></p><p>
Here are some properties of definite integrals. We have discussed and used most of these properties for [mathjaxinline]f>0[/mathjaxinline]. The properties below are true for definite integrals for functions which can be negative as well. </p><dl class="description"><dt>Sums:</dt><dd><table id="a0000000977" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{a}^{b} \left( f(x) +g(x) \right) \, dx \, =\, \int _{a}^{b} f(x)\, dx +\int _{a}^{b} g(x) \, dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>Constant Multiples:</dt><dd><table id="a0000000978" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{a}^{b} c \, f(x) \, dx \, =\, c\, \int _{a}^{b} f(x)\, dx\qquad \text {for any constant}\, c[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>Same Upper and Lower Limits:</dt><dd><table id="a0000000979" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{a}^{a} \, f(x) \, dx\, =\, 0[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>Reversing Limits of Integrals:</dt><dd><table id="a0000000980" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{b}^{a} \, f(x) \, dx\, =\, - \int _{a}^{b} \, f(x) \, dx\qquad \text {for any}\, a,b[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This is the definition of a definite integral with its lower limit greater than its upper limit. It is defined this way to be consistent with FTC1. </p></dd><dt>Combining integrals:</dt><dd><table id="a0000000981" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{a}^{c} \, f(x) \, dx\, =\, \int _{a}^{b} \, f(x) \, dx\, +\, \int _{b}^{c} \, f(x) \, dx\qquad \text {for any}\, a,b,c[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Notice that we can use this property with [mathjaxinline]a[/mathjaxinline],[mathjaxinline]b[/mathjaxinline],[mathjaxinline]c[/mathjaxinline] in any order. </p></dd></dl>
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<h2 class="hd hd-2 unit-title">12. Using the properties of integrals</h2>
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<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Using integral properties 2
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{1}^{0} \sqrt {1-t^2} \, \, dt=[/mathjaxinline]</p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Using integral properties 3
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Suppose [mathjaxinline]\, \displaystyle \int _{-1}^{3} f(u) \, du\, =\, A,\, \,[/mathjaxinline] and [mathjaxinline]\, \displaystyle \int _{1}^{3} f(u) \, du\, =\, B[/mathjaxinline].<br/><p style="display:inline">Then, [mathjaxinline]\displaystyle \int _{-1}^{1} f(u)\, du=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory2-tab12-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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(Enter your answers in terms of [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline].)<br/></p>
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<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Using integral properties 4
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{3}^{9\pi /2} \sin (n\theta )\, d\theta + \int _{9\pi /2}^{3+2\pi /n} \sin (n\theta )\, d\theta =[/mathjaxinline]</p>
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<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Even functions
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Let [mathjaxinline]f(x)[/mathjaxinline] be an even function and [mathjaxinline]b&gt;0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^{b} f(x) \, dx \, =\, c \left(\int _{0}^{-b} f(x) \, dx \, \right)\,[/mathjaxinline] for [mathjaxinline]\, \, c\, =\,[/mathjaxinline]</p>
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Odd functions
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Let [mathjaxinline]\, g(t)[/mathjaxinline] be an odd function and [mathjaxinline]b&gt;0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^{b} g(t)\, dt = c \, \left(\int _{0}^{-b} g(t)\, dt \right)\, \,[/mathjaxinline] for [mathjaxinline]\, \, c\, =\,[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">13. Estimation</h2>
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If [mathjaxinline]\, \, f(x)\leq g(x)[/mathjaxinline], and [mathjaxinline]\, a\leq b,\,[/mathjaxinline]then </p><table id="a0000001001" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x)\, dx\, \, \, \leq \, \, \, \int _{a}^{b} g(x)\, dx \qquad (\text {for}\, \, a\leq b).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Notice that the order of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] matters for this inequality. If [mathjaxinline]b \leq a[/mathjaxinline], that is, the order of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] are reversed, then since </p><table id="a0000001002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle -\int _{a}^{b} f(x)\, dx\, \, \, = \, \, \, \int _{b}^{a} f(x)\, dx,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and the second integral has the lower limit smaller than the upper limit, we can use the inequality above on the second integral.<br/>This gives </p><table id="a0000001003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]-\left(\int _{a}^{b} f(x)\, dx\right)\, \, \, = \, \, \, \int _{b}^{a} f(x)\, dx\, \, \, \leq \, \, \, \int _{b}^{a} g(x)\, dx\, \, \, =\, \, \, -\left( \int _{a}^{b} g(x)\, dx \right)\qquad (b\leq a).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Multiplying this inequality by [mathjaxinline]-1[/mathjaxinline] and omitting two integrals in the middle, we get </p><table id="a0000001004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x)\, dx\, \geq \, \int _{a}^{b} g(x)\, dx \qquad (b\leq a).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In other words, if the limits of the integrals are reversed, the inequality is also reversed. </p>
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<h2 class="hd hd-2 unit-title">14. Inequalities</h2>
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Inequalities of the Logarithm using integrals
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We will integrate an algebraic inequality to get an inequality on the logarithmic function.<br/></p>
<p>
Given the following factorization. </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001005" style="table-layout:auto" width="100%">
<tr id="a0000001006">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle x^{n+1}-1[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle -(1-x)\left(1+x+x^2+\cdots +x^{n-1}+x^ n\right)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.154)</td>
</tr>
<tr id="a0000001007">
<td style="width:40%; border:none">&#160;</td>
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&#160;
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle -(1-x)\, \left(1+ \sum _{k=1}^{n} x^ k\right)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.155)</td>
</tr>
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<p>
Use the above equality to find fractional expression for the sum below. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{1}{1-x}- \left(1+ \sum _{k=1}^{n} x^ k\right)\, =\,[/mathjaxinline]</p>
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Therefore, for any [mathjaxinline]0\leq x&lt;1[/mathjaxinline] and any integer [mathjaxinline]n\geq 0[/mathjaxinline], which of the following inequalities can you conclude? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory2-tab14-problem1_3_1">
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<input type="radio" name="input_theory2-tab14-problem1_3_1" id="input_theory2-tab14-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory2-tab14-problem1_3_1-choice_1-label" for="input_theory2-tab14-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory2-tab14-problem1_3_1"> <text> [mathjaxinline]\displaystyle \frac{1}{1-x}\, \, &gt;\, \, 1+ \sum _{k=1}^{n} x^ k[/mathjaxinline]</text>
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<input type="radio" name="input_theory2-tab14-problem1_3_1" id="input_theory2-tab14-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory2-tab14-problem1_3_1-choice_2-label" for="input_theory2-tab14-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory2-tab14-problem1_3_1"> <text> [mathjaxinline]\displaystyle \frac{1}{1-x}\, \, \geq \, \, 1+ \sum _{k=1}^{n} x^ k[/mathjaxinline]</text>
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<input type="radio" name="input_theory2-tab14-problem1_3_1" id="input_theory2-tab14-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory2-tab14-problem1_3_1-choice_3-label" for="input_theory2-tab14-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory2-tab14-problem1_3_1"> <text> [mathjaxinline]\displaystyle \frac{1}{1-x}\, \, &lt; \, \, 1+ \sum _{k=1}^{n} x^ k[/mathjaxinline]</text>
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<input type="radio" name="input_theory2-tab14-problem1_3_1" id="input_theory2-tab14-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="theory2-tab14-problem1_3_1-choice_4-label" for="input_theory2-tab14-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory2-tab14-problem1_3_1"> <text> [mathjaxinline]\displaystyle \frac{1}{1-x}\, \, \leq \, \, 1+ \sum _{k=1}^{n} x^ k[/mathjaxinline]</text>
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Finally, integrate the inequality of [mathjaxinline]\displaystyle \frac{1}{1-x}[/mathjaxinline] from [mathjaxinline]x=0[/mathjaxinline] to [mathjaxinline]x=b[/mathjaxinline] to get an inequality of the logarithmic function.<br/>For any [mathjaxinline]\, \, 0\leq b&lt;1[/mathjaxinline] and any integer [mathjaxinline]\, \, n\geq 0[/mathjaxinline], which of the following inequalities of the logarithm function is true ?<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory2-tab14-problem1_4_1">
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<input type="radio" name="input_theory2-tab14-problem1_4_1" id="input_theory2-tab14-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory2-tab14-problem1_4_1-choice_1-label" for="input_theory2-tab14-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory2-tab14-problem1_4_1"> <text> [mathjaxinline]\displaystyle \ln (1-b)&lt;-\sum _{k=1}^{n+1} b^ k[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_theory2-tab14-problem1_4_1" id="input_theory2-tab14-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory2-tab14-problem1_4_1-choice_2-label" for="input_theory2-tab14-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory2-tab14-problem1_4_1"> <text> [mathjaxinline]\displaystyle \ln (1-b)&gt; 1+\sum _{k=1}^{n+1} b^ k[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_theory2-tab14-problem1_4_1" id="input_theory2-tab14-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory2-tab14-problem1_4_1-choice_3-label" for="input_theory2-tab14-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory2-tab14-problem1_4_1"> <text> [mathjaxinline]\displaystyle \ln (1-b)\geq 1+\sum _{k=1}^{n+1} \frac{b^ k}{k}[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_theory2-tab14-problem1_4_1" id="input_theory2-tab14-problem1_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="theory2-tab14-problem1_4_1-choice_4-label" for="input_theory2-tab14-problem1_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory2-tab14-problem1_4_1"> <text> [mathjaxinline]\displaystyle \ln (1-b)\leq \frac{1}{n}\left(-1-\sum _{k=1}^{n+1} b^ k\right)[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">15. Change of variables for definite integrals</h2>
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<p><b class="bfseries">Change of variables for definite integrals</b></p><p>
When we make a change of variables of an integral in order to evaluate it, we are using the method of substitution. The method of substitution for definite integrals is exactly analogous to the method of substitution for indefinite integrals, except we now need to pay attention to the limits of the integrals. </p><p>
If </p><table id="a0000001018" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x)\, dx = \int _{a}^{b} g(u(x)) u'(x) \, dx,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and [mathjaxinline]u'[/mathjaxinline] does not change sign between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline], then </p><table id="a0000001019" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x)\, dx = \int _{a}^{b} g(u(x)) u'(x) \, dx = \int _{u(a)}^{u(b)} g(u) \, du.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
That is, the limits of the integral over [mathjaxinline]u[/mathjaxinline] are the values of [mathjaxinline]u[/mathjaxinline] corresponding to the limits of the integral over [mathjaxinline]x[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">16. Practice with substitution</h2>
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For the problems below, you may enter your numerical answers as formulas evaluated at different values. For example, enter arctan(2) for the number [mathjaxinline]\arctan (2)[/mathjaxinline]. You may also enter numerical values, especially for formulas which are easy to evaluate. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{\sqrt {2}}^{2} \, x \sqrt {4x^2-7}\, dx=[/mathjaxinline]</p>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Substitution practice 2
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^{\pi /3} \tan ^2(\beta ) \sec ^2(\beta )\, d\beta \, =\,[/mathjaxinline]</p>
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<td class="formulainput">Integers</td>
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<font color="#0078b0">2520</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Substitution practice 3
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-4}^{-3} \frac{1}{t+2} \, dt=[/mathjaxinline]</p>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Substitution practice 4
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-2}^{0} \frac{1}{-y^3+3y^2-3y+1} \, dy=[/mathjaxinline]</p>
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(Hint: First rewrite the integrand.)<br/></p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Substitution practice 5
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-\pi /2}^{-\pi /4} \tan \left(\phi +\frac{\pi }{2}\right)\, d\phi =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">17. Caution for the method of substitution</h2>
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<p><b class="bfseries">Caution for the method of substitution</b></p><p>
When we use the method of substitution, we need to be very careful about when [mathjaxinline]u'[/mathjaxinline] (or [mathjaxinline]du[/mathjaxinline]) changes sign. If [mathjaxinline]u'[/mathjaxinline] changes sign within the integration interval [mathjaxinline][a,b][/mathjaxinline], the method of substitution may give the wrong answer. In this case, we need to first rewrite the integral as a sum of two integrals such that within the limits of each integral [mathjaxinline]u'[/mathjaxinline] does not change sign, and then use the method of substitution on each integral separately.<br/></p><p><b class="bfseries">Example: [mathjaxinline]\displaystyle \int _{-1}^{1} x^2\, dx[/mathjaxinline] </b></p><p>
Here is an example to show what goes wrong and how to use the method of substitution correctly if [mathjaxinline]u'[/mathjaxinline] changes sign within the limits of the integral. It is just for illustrative purpose because we can easily compute this integral without using the method of substitution. </p><p>
First, [mathjaxinline]\displaystyle \int _{-1}^{1} x^2 \, dx \neq 0[/mathjaxinline] by direct computation using FTC1 or by the fact that the area under [mathjaxinline]y=x^2[/mathjaxinline] between [mathjaxinline]-1[/mathjaxinline] and [mathjaxinline]1[/mathjaxinline] is non-zero.<br/></p><p>
On the other hand, if we use the method of substitution and let [mathjaxinline]u=x^2[/mathjaxinline], then [mathjaxinline]du=2x\, dx[/mathjaxinline]. Notice [mathjaxinline]du[/mathjaxinline] changes sign at [mathjaxinline]x=0[/mathjaxinline], which is between the lower and upper limits of the integral. Now, if we make the mistake of applying substitution directly, we get </p><table id="a0000001028" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{-1}^{1} x^2 \, dx = \int _{u=(-1)^2}^{u=1^2} u \cdot \frac{du}{2\sqrt {u}} = \int _{1}^{1} u \cdot \frac{du}{2\sqrt {u}}=0[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This is clearly incorrect.<br/></p><p><b class="bfseries">Using substitution correctly</b></p><p>
To use the substitution [mathjaxinline]u=x^2[/mathjaxinline] correctly, we need to first break the integral into two pieces so that [mathjaxinline]u'[/mathjaxinline] does not change sign within the limits of each of the two integral. Since [mathjaxinline]u'[/mathjaxinline] changes sign at [mathjaxinline]0[/mathjaxinline], we will break the integral into two at [mathjaxinline]0[/mathjaxinline]. </p><table id="a0000001029" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001030"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _{-1}^{1} x^2 \, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \int _{-1}^{0} x^2 \, dx + \int _{0}^{1} x^2 \, dx.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.167)</td></tr></table><p>
Then, since [mathjaxinline]x^2[/mathjaxinline] is even, [mathjaxinline]\displaystyle \int _{-1}^{1} x^2 \, dx =2 \int _{0}^{1} x^2 \, dx[/mathjaxinline]. Since [mathjaxinline]u'[/mathjaxinline] does not change sign within [mathjaxinline][0,1][/mathjaxinline], we can use the method of substitution with [mathjaxinline]u=x^2[/mathjaxinline] on [mathjaxinline]\displaystyle \int _{0}^{1} x^2 \, dx[/mathjaxinline]. </p><table id="a0000001031" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\left.\int _{0}^{1} x^2 \, dx = \int _{(0)^2}^{1^2} u \cdot \frac{du}{2\sqrt {u}} = \frac{1}{2}\left(\frac{2}{3}u^{\frac{3}{2}}\right) \right|_{0}^{1}= \frac{1}{3}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This gives [mathjaxinline]\displaystyle \int _{-1}^{1} x^2 \, dx =\frac{2}{3}[/mathjaxinline], which is the correct answer.<br/></p><p><b class="bfseries">Explanation</b></p><p>
So what went wrong when we apply this substitution [mathjaxinline]\displaystyle \int _{-1}^{1} x^2 \, dx[/mathjaxinline]? What happens is that when [mathjaxinline]u'[/mathjaxinline] changes sign, [mathjaxinline]u[/mathjaxinline] must be sometimes increasing and sometimes decreasing. But this means that [mathjaxinline]u(x)[/mathjaxinline] does not have a well-defined inverse. That is, there is not one formula [mathjaxinline]x=x(u)[/mathjaxinline] that works in the whole interval of integration. In this example, </p><table id="a0000001032" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle x\, \, = \, \, \begin{cases} \displaystyle +\sqrt {u} & \mbox{if } x \geq 0 \\ -\sqrt {u} & \mbox{if } x\leq 0. \end{cases}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(2.168)</td></tr></table><p>
If we first break the region of integration into regions for which the inverse of [mathjaxinline]u[/mathjaxinline] is well-defined, the method of substitution can be used in each region. </p>
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<h2 class="hd hd-2 unit-title">18. Problems on method of substitution</h2>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Evaluate the integral of the same function as above but with the lower limit changed. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-2}^{1} \sqrt {(x^2+1)^2-1} \, \, dx\, =\,[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">19. Comparing FTC1 with MVT</h2>
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Review of MVT
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We will end this section with a comparison between the First Fundamental Theorem of Calculus and the Mean Value Theorem. </p>
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Let [mathjaxinline]F(x)[/mathjaxinline] be differentiable on [mathjaxinline][a,b][/mathjaxinline]. And let </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001059" style="table-layout:auto" width="100%">
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[mathjaxinline]\displaystyle \Delta F[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F(b)-F(a)[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.188)</td>
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[mathjaxinline]\displaystyle \Delta x[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle b-a.[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.189)</td>
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The MVT implies that [mathjaxinline]\displaystyle \frac{\Delta F}{\Delta x}[/mathjaxinline] equals <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory2-tab19-problem1_2_1">
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<input type="radio" name="input_theory2-tab19-problem1_2_1" id="input_theory2-tab19-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory2-tab19-problem1_2_1-choice_2-label" for="input_theory2-tab19-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory2-tab19-problem1_2_1"> <text> [mathjaxinline]\displaystyle \frac{1}{b-a} \int _ a^ b F'(x) \, dx[/mathjaxinline]</text>
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Let [mathjaxinline]F(x)[/mathjaxinline] be differentiable and [mathjaxinline]F'(x)[/mathjaxinline] be continuous on [mathjaxinline][a,b][/mathjaxinline]. And as above, denote </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001064" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta F[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F(b)-F(a)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.191)</td>
</tr>
<tr id="a0000001066">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta x[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle b-a.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.192)</td>
</tr>
</table>
<p>
The FTC1 implies that [mathjaxinline]\displaystyle \frac{\Delta F}{\Delta x}[/mathjaxinline] equals<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory2-tab19-problem2_2_1">
<fieldset aria-describedby="status_theory2-tab19-problem2_2_1">
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<input type="radio" name="input_theory2-tab19-problem2_2_1" id="input_theory2-tab19-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory2-tab19-problem2_2_1-choice_1-label" for="input_theory2-tab19-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory2-tab19-problem2_2_1"> <text> [mathjaxinline]F'(c)\,[/mathjaxinline] for some [mathjaxinline]\, c,\, \, a&lt;c&lt;b[/mathjaxinline]</text>
</label>
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<div class="field">
<input type="radio" name="input_theory2-tab19-problem2_2_1" id="input_theory2-tab19-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory2-tab19-problem2_2_1-choice_2-label" for="input_theory2-tab19-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory2-tab19-problem2_2_1"> <text> [mathjaxinline]\displaystyle \frac{1}{b-a} \int _ a^ b F'(x) \, dx[/mathjaxinline]</text>
</label>
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<div class="field">
<input type="radio" name="input_theory2-tab19-problem2_2_1" id="input_theory2-tab19-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory2-tab19-problem2_2_1-choice_3-label" for="input_theory2-tab19-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory2-tab19-problem2_2_1"> <text> [mathjaxinline]\displaystyle \int _ a^ b F'(x) \, dx[/mathjaxinline]</text>
</label>
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<h2 class="hd hd-2 unit-title">20. FTC1 versus MVT</h2>
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<p>
Let us compare what the First Fundamental Theorem of Calculus and what the Mean Value Theorem say about the average rate of change of a function.<br/></p><p>
If [mathjaxinline]F(x)[/mathjaxinline] is differentiable and [mathjaxinline]F'(x)[/mathjaxinline] is continuous on [mathjaxinline]\, [a,b][/mathjaxinline]. And let </p><table id="a0000001071" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001072"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta F[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F(b)-F(a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.195)</td></tr><tr id="a0000001073"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle b-a.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.196)</td></tr></table><p>
Then, the MVT states that </p><table id="a0000001074" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001075"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{\Delta F}{\Delta x}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F'(c) \qquad \text {for some}\, c,\, \, a<c<b.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.197)</td></tr></table><p>
On the other hand, the FTC1 gives </p><table id="a0000001076" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001077"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{\Delta F}{\Delta x}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{b-a} \int _ a^ b F'(x) \, dx.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.198)</td></tr></table><p>
We see that the FTC1 gives a specific value for [mathjaxinline]\displaystyle \frac{\Delta F}{\Delta x}[/mathjaxinline], the average rate of change of [mathjaxinline]F[/mathjaxinline] over [mathjaxinline][a,b][/mathjaxinline], but the MVT does not, since it does not tell us where [mathjaxinline]c[/mathjaxinline] is.<br/></p><p>
Therefore, the First Fundamental Theorem is much more useful than the Mean Value Theorem. Once we have FTC1 at our disposal, we do not need to use MVT anymore. Nonetheless, the Mean Value Theorem is important as the basis of calculus. We needed it to establish the fact that two antiderivatives of the same function can only differ by a constant. We will need this fact again in order to finally prove FTC1 along with FTC2 in the next section.<br/></p>
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<h2 class="hd hd-2 unit-title">21. Summary</h2>
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<p><b class="bfseries">The First Fundamental Theorem of Calculus</b></p><p>
The <span style="color:#99182C"><b class="bf">First Fundamental Theorem of Calculus</b></span> states that:<br/>If [mathjaxinline]F[/mathjaxinline] is differentiable, and [mathjaxinline]F'=f[/mathjaxinline] is continuous, then </p><table id="a0000001078" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001079"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(b)-F(a)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left.\phantom{\int } F(x)\, \right|_ a^ b.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.199)</td></tr></table><p>
In other words, the definite integral of a function is the difference between the values of its antiderivative at the limits of the definite integral.<br/></p><p>
We will abbreviate the <span style="color:#99182C"><b class="bf">First Fundamental Theorem of Calculus</b></span> as <span style="color:#99182C"><b class="bf">FTC1</b></span>.<br/></p><p>
The FTC1 connects the definite integral to the antiderivative. With this connection, we can now compute definite integrals using antiderivatives, and dispense with Riemann sums. </p><p><b class="bfseries">The definite integral of any continuous function</b></p><p>
Since the equation in the statement of FTC1 makes sense for general functions, we can use the FTC1 to extend the <span style="color:#27408C"><b class="bf">definition of the definite integral</b></span> to functions that are not necessarily non-negative. That is,<br/>For any continuous function [mathjaxinline]f[/mathjaxinline] with an antiderivative [mathjaxinline]F[/mathjaxinline], </p><table id="a0000001080" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001081"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x) \, \, \, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F(b)-F(a)\qquad (\text {for any continuous}\, \, \, f).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.200)</td></tr></table><p>
It turns out that to be consistent with FTC1, the Riemann sum formula for definite integrals does not need to change. Consequently, the <span style="color:#27408C"><b class="bf">geometric definition</b></span> of [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline] that is consistent with FTC1 is as follows. </p><center><img src="/assets/courseware/v1/2ebc4ef59dc3ea3cc671f1d22d6de4f3/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc1_signedarea.svg" width="300px" alt=" " style="margin: 10px 25px 25px 25px"/><br/></center><table id="a0000001082" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x) \, dx\, =\, \text {Area above} \, x\text {-axis and below }\, y=f(x)\, -\, \text {Area below} \, x\text {-axis and above }\, y=f(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where the areas considered are between the vertical lines [mathjaxinline]x=a[/mathjaxinline] and [mathjaxinline]x=b[/mathjaxinline]. The important point is that the area above the [mathjaxinline]x[/mathjaxinline]-axis is counted with a positive sign and the area below the [mathjaxinline]x[/mathjaxinline]-axis is counted with a negative sign. In other words, the definite integral of a general function is the <span style="color:#99182C"><b class="bf">signed area bounded by the curve [mathjaxinline]y=f(x)[/mathjaxinline]</b></span>.<br/></p><p>
We will continue to use FTC1, not Riemann sums, to evaluate definite integrals. We will also often use the area interpretation to deduce properties of definite integrals, for example when looking for symmetry. </p><p><b class="bfseries">Velocity and speed, displacement and distance travelled</b></p><p>
Suppose you are travelling between time [mathjaxinline]a[/mathjaxinline] and time [mathjaxinline]b[/mathjaxinline], and your velocity is given by [mathjaxinline]v(t)[/mathjaxinline] and your position by [mathjaxinline]x(t)[/mathjaxinline] at time [mathjaxinline]t[/mathjaxinline].<br/></p><p>
Recall that <span style="color:#27408C"><b class="bf">Speed</b></span> is [mathjaxinline]|v(t)|[/mathjaxinline], the absolute value of the velocity function.<br/></p><p>
Then, </p><table id="a0000001083" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001084"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _{a}^{b} v(t) \, dt[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(b)-x(a) \qquad (\text {Displacement}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.201)</td></tr></table><p>
That is, the integral of the velocity function is the (net) change of position between time [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline], also called the <span style="color:#27408C"><b class="bf">displacement</b></span>.<br/></p><p>
On the other hand, [mathjaxinline]\, \, \displaystyle \int _{a}^{b} \left|v(t)\right| \, \, dt\, \,[/mathjaxinline] gives the <span style="color:#27408C"><b class="bf">total distance travelled</b></span> between time [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p><p>
In the case when [mathjaxinline]v(t)>0[/mathjaxinline] for all [mathjaxinline]t[/mathjaxinline], [mathjaxinline]|v(t)|=v(t)[/mathjaxinline], and therefore [mathjaxinline]\, \, \displaystyle \int _{a}^{b} v(t) \, dt \, \, =\, \, \int _{a}^{b} \left|v(t)\right| \, \, dt.\, \, \, \,[/mathjaxinline] In other words, when you are always travelling in the same direction, then speed is equal to velocity, and total distance traveled is equal to displacement. </p><p><b class="bfseries">Properties of definite integrals</b></p><p>
Here are some properties of definite integrals. We have discussed and used most of these properties for [mathjaxinline]f>0[/mathjaxinline]. The properties below are true for definite integrals for functions which can be negative as well. </p><dl class="description"><dt>Sums:</dt><dd><table id="a0000001085" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{a}^{b}\left( f(x) +g(x) \right) \, dx \, =\, \int _{a}^{b} f(x)\, dx +\int _{a}^{b} g(x) \, dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>Constant Multiples:</dt><dd><table id="a0000001086" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{a}^{b} c \, f(x) \, dx \, =\, c\, \int _{a}^{b} f(x)\, dx\qquad \text {for any constant}\, c[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>Same Upper and Lower Limits:</dt><dd><table id="a0000001087" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{a}^{a} \, f(x) \, dx\, =\, 0[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>Reversing Limits of Integrals:</dt><dd><table id="a0000001088" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{b}^{a} \, f(x) \, dx\, =\, - \int _{a}^{b} \, f(x) \, dx\qquad \text {for any}\, a,b[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This is the definition of a definite integral with its lower limit greater than its upper limit. It is defined this way to be consistent with FTC1. </p></dd><dt>Combining integrals:</dt><dd><table id="a0000001089" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{a}^{c} \, f(x) \, dx\, =\, \int _{a}^{b} \, f(x) \, dx\, +\, \int _{b}^{c} \, f(x) \, dx\qquad \text {for any}\, a,b,c[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
We have seen this property in the last section with [mathjaxinline]a<b<c[/mathjaxinline]. With the property above on reversing the limits of a definite integral, we can now use this property with [mathjaxinline]a[/mathjaxinline],[mathjaxinline]b[/mathjaxinline],[mathjaxinline]c[/mathjaxinline] in any order. </p></dd></dl><p><b class="bfseries">Estimation</b></p><p>
If [mathjaxinline]\, f(x)\leq g(x)[/mathjaxinline], and [mathjaxinline]\, a\leq b,\,[/mathjaxinline] then </p><table id="a0000001090" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x)\, dx\, \, \, \leq \, \, \, \int _{a}^{b} g(x)\, dx \qquad (\text {for}\, \, a\leq b).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Notice that the order of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] matters for this inequality. If instead of [mathjaxinline]a\leq b[/mathjaxinline] we have [mathjaxinline]b\leq a[/mathjaxinline], then </p><table id="a0000001091" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x)\, dx\, \geq \, \int _{a}^{b} g(x)\, dx \qquad (b\leq a).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In other words, if the limits of the integrals are reversed, the inequality is also reversed. </p><p><b class="bfseries">Change of variables for definite integrals</b></p><p>
When we make a change of variables of an integral in order to evaluate it, we are using the method of substitution. The method of substitution for definite integrals is exactly analogous to the method of substitution for indefinite integrals, except we now need to pay attention to the limits of the integrals. </p><p>
If </p><table id="a0000001092" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x)\, dx = \int _{a}^{b} g(u(x)) u'(x) \, dx,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and [mathjaxinline]u'[/mathjaxinline] does not change sign between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline], then </p><table id="a0000001093" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x)\, dx = \int _{a}^{b} g(u(x)) u'(x) \, dx = \int _{u(a)}^{u(b)} g(u) \, du.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
That is, the limits of the integral over [mathjaxinline]u[/mathjaxinline] are the values of [mathjaxinline]u[/mathjaxinline] corresponding to the limits of the integral over [mathjaxinline]x[/mathjaxinline]. </p><p><b class="bfseries">Caution</b></p><p>
When we use the method of substitution, we need to be very careful about when [mathjaxinline]u'[/mathjaxinline] (or [mathjaxinline]du[/mathjaxinline]) changes sign. If [mathjaxinline]u'[/mathjaxinline] changes sign within the integration interval [mathjaxinline][a,b][/mathjaxinline], the method of substitution may give the wrong answer. In this case, we need to first rewrite the integral as a sum of two integrals such that within the limits of each integral [mathjaxinline]u'[/mathjaxinline] does not change sign, and then use the method of substitution on each integral separately.<br/></p><p><b class="bfseries">Comparing FTC1 and MVT</b></p><p>
If [mathjaxinline]F(x)[/mathjaxinline] is differentiable and [mathjaxinline]F'(x)[/mathjaxinline] is continuous on [mathjaxinline]\, [a,b][/mathjaxinline]. And let </p><table id="a0000001094" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001095"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta F[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F(b)-F(a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.202)</td></tr><tr id="a0000001096"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle b-a.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.203)</td></tr></table><p>
Then, the MVT states that </p><table id="a0000001097" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001098"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{\Delta F}{\Delta x}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F'(c) \qquad \text {for some}\, c,\, \, a<c<b.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.204)</td></tr></table><p>
On the other hand, the FTC1 gives </p><table id="a0000001099" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001100"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{\Delta F}{\Delta x}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{b-a} \int _ a^ b F'(x) \, dx.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.205)</td></tr></table><p>
We see that the FTC1 gives a specific value for [mathjaxinline]\displaystyle \frac{\Delta F}{\Delta x}[/mathjaxinline], the average rate of change of [mathjaxinline]F[/mathjaxinline] over [mathjaxinline][a,b][/mathjaxinline], but the MVT does not, since it does not tell us where [mathjaxinline]c[/mathjaxinline] is.<br/></p><p>
Therefore, the First Fundamental Theorem much more useful than the Mean Value Theorem. Once we have FTC1 at our disposal, we do not need to use MVT anymore. Nonetheless, the Mean Value Theorem is important as the basis of calculus. We needed it to establish the fact that two antiderivatives of the same function can only differ by a constant. We will need this fact again in order to prove FTC1 in the next section.<br/></p>
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