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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">The definite integral</h3>
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<h2 class="hd hd-2 unit-title">2. The definite integral</h2>
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<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Know the geometric definition of a <span style="color:#27408C"><b class="bf">definite integral</b></span> as the <span style="color:#27408C"><b class="bf">area under a curve</b></span>. </p></li><li><p>
Evaluate simple definite integrals. </p></li><li><p>
Use the <span style="color:#27408C"><b class="bf">[mathjaxinline]\displaystyle \sum[/mathjaxinline] notation</b></span> for sums. </p></li><li><p>
Approximate a definite integral using a <span style="color:#27408C"><b class="bf">Riemann Sum</b></span>. </p></li></ul><p><b class="bfseries">Contents: 17 pages</b></p><p>
15 videos (80 minutes 1x speed) 35 questions </p>
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<h2 class="hd hd-2 unit-title">3. The geometric definition of the definite integral</h2>
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<h3 class="hd hd-2">Definition and first examples</h3>
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<p><b class="bfseries">The geometric definition of the definite integral </b></p><p>
We will only deal with definite integrals of non-negative functions in this section.<br/></p><p>
Geometrically, the <span style="color:#99182C"><b class="bf">definite integral of [mathjaxinline]f\,[/mathjaxinline] from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]b[/mathjaxinline]</b></span>, denoted by </p><table id="a0000000723" 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,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
is the area under the graph of [mathjaxinline]f(x)[/mathjaxinline] between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/b4e1e194803ea64ed5decb84891c8a57/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_riemann4.svg" width="250px" alt="A function y equals f of x is plotted in the first quadrant. The points a and b are indicated on the x axis. The region beneath the function and between the vertical lines x equals a and x equals b is shaded." style="margin: 10px 25px 25px 25px"/></center><p>
More precisely, this definite integral is equal to the area of the region above the [mathjaxinline]x[/mathjaxinline]-axis, below the curve [mathjaxinline]y=f(x)[/mathjaxinline], and in between the two vertical lines [mathjaxinline]x=a[/mathjaxinline] and [mathjaxinline]x=b[/mathjaxinline], as shown shaded in the figure below. </p><p>
The [mathjaxinline]x[/mathjaxinline]-values [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] are called the <span style="color:#99182C"><b class="bf">lower and upper limits of the integral</b></span>. (This is a different sense of the word "limit" from when we take a limit of a function.) <br/></p><p>
The only difference between the notation for definite and indefinite integrals is that definite integrals have limits but indefinite integrals do not.<br/></p>
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Given the graph of the function [mathjaxinline]y=e^ x[/mathjaxinline] below. </p>
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Write the area of the shaded region as a definite integral.<br/></p>
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(<i class="itshape">Enter the upper limit in the answer box above the integral symbol</i> [mathjaxinline]\int[/mathjaxinline], <i class="itshape">the lower limit in the answer box below the integral symbol, and the integrand to the right of the integral symbol. Note the term</i> [mathjaxinline]dx[/mathjaxinline] <i class="itshape">is provided for you, so you need not include it in your integrand.</i> Note that all 3 answer boxes are graded together.) </p>
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<span class="trailing_text" id="trailing_text_theory1-tab3-problem1_2_2"> [mathjaxinline] dx [/mathjaxinline]</span>
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Throughout this section, there will be problems that guide you through the evaluation of this definite integral. </p>
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<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/>
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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">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>
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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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<h2 class="hd hd-2 unit-title">4. First examples of definite integrals</h2>
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Simple integral 1
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Let us compute some definite integrals of the constant function [mathjaxinline]f(x)=2[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _1^4 2 \, dx=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-1}^4 2 \, dx=[/mathjaxinline]</p>
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For [mathjaxinline]b\geq 0[/mathjaxinline], <p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^ b x^0 \, dx=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory1-tab4-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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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 \( 2\pi \)
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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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Simple integrals 3
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<p style="display:inline">[mathjaxinline]\displaystyle \int _0^2 x \, dx=[/mathjaxinline]</p>
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<p style="display:inline">For any [mathjaxinline]b\geq 0[/mathjaxinline], [mathjaxinline]\displaystyle \int _0^ b x \, dx=[/mathjaxinline]</p>
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<p style="display:inline">If [mathjaxinline]b\geq 2[/mathjaxinline], [mathjaxinline]\displaystyle \int _2^ b 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">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>
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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>
</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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Combining integrals
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Given any [mathjaxinline]f(x)[/mathjaxinline] (non-negative), and [mathjaxinline]a&lt;b&lt;c[/mathjaxinline]. </p>
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If [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx\, =\, 50[/mathjaxinline], and [mathjaxinline]\displaystyle \int _ b^ c f(x) \, dx\, =\, 20[/mathjaxinline], then <p style="display:inline">[mathjaxinline]\displaystyle \int _ a^ c f(x) \, dx \, =[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_theory1-tab4-problem4_2_1" class="text-input-dynamath capa_inputtype inline textline">
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(Enter the integration limits above and below the last integration symbol [mathjaxinline]\int[/mathjaxinline].)<br/></p>
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In general, <span><style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
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</style><table><tbody><tr><td/><td><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_theory1-tab4-problem4_3_1" class=" capa_inputtype textline">
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\( \displaystyle \large{ \int_a^b f(x) \, dx \,+ \, \int_b^c f(x) \, dx \quad =} \)
</p></td><td><p style="display:inline; text-align:right"> \( \displaystyle \large{ \int f(x)\, dx } \)</p></td></tr><tr><td/><td><div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_theory1-tab4-problem4_4_1" class=" capa_inputtype textline">
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<h2 class="hd hd-2 unit-title">5. Even symmetry</h2>
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Simple integrals 4
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-3}^3 \left|x\right| \, dx=[/mathjaxinline]</p>
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Simple integrals 5
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(Hint: draw a graph.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-3}^3 \, \sqrt {9-x^2} \, dx=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _0^3 \, \sqrt {9-x^2} \, dx=[/mathjaxinline]</p>
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Symmetry of definite integrals
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Suppose [mathjaxinline]f(x)[/mathjaxinline] is even. Recall a function [mathjaxinline]f(x)[/mathjaxinline] is even if [mathjaxinline]f(-x)=f(x)[/mathjaxinline].<br/></p>
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<p style="display:inline">If [mathjaxinline]\displaystyle \, \int _{0}^ b f(x) \, dx =6[/mathjaxinline], then [mathjaxinline]\displaystyle \, \int _{-b}^ b f(x) \, dx =[/mathjaxinline]</p>
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<p style="display:inline">Hence, for any even function [mathjaxinline]f(x)[/mathjaxinline], and [mathjaxinline]b&gt;0[/mathjaxinline],<br/>[mathjaxinline]\displaystyle \, \int _{0}^ b f(x) \, dx = c \, \int _{-b}^ b f(x) \, dx[/mathjaxinline], where [mathjaxinline]c=[/mathjaxinline]</p>
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Piecewise linear functions
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Given the graph of [mathjaxinline]y=f(x)[/mathjaxinline] below. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _0^3 \, f(x)\, dx=[/mathjaxinline]</p>
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By only looking at the graph of [mathjaxinline]y=x^2[/mathjaxinline], decide which of the following are greater than [mathjaxinline]\displaystyle \int _0^ b \, x^2 \, dx\,[/mathjaxinline] for every [mathjaxinline]b&gt;0[/mathjaxinline]? (Check all that apply.)<br/></p>
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We will now discuss how to find the area under the graph of a general function [mathjaxinline]f(x)[/mathjaxinline] between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/b4e1e194803ea64ed5decb84891c8a57/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_riemann4.svg" width="250px" alt="A function y equals f of x is plotted in the first quadrant. The points a and b are indicated on the x axis. The region beneath the function and between the vertical lines x equals a and x equals b is shaded." style="margin: 10px 25px 25px 25px"/></center>
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<p><b class="bfseries">Main idea</b></p><p>
In general, the region under a curve is not a simple shape whose area we have a formula for. In this case, we will start with an approximation of the area and then take the limit as the approximation approaches the actual area. We will do this in the following three steps. </p><ol class="enumerate"><li value="1"><p>
Divide the region under the curve into “strips", as shown in the figure below. </p><center><img src="/assets/courseware/v1/b40949e8868dfa3ff56e8e6d1907be39/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_riemann1decreasing.svg" width="250px" alt="A function y equals f of x is plotted in the first quadrant. The vertical lines x equals a and x equals b are indicated by dashed lines. Three additional dashed lines that start at the x axis and extend upwards to the function height are indicated by dashed lines between x equals a and x equals b." style="margin: 10px 25px 25px 25px"/></center><p>
Approximate the area of each strip by the area of a rectangle. </p><center><img src="/assets/courseware/v1/a9cbe9fbbcd30c1afbb09fb0cf5a0539/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_riemann2decreasing.svg" width="250px" alt="A function y equals f of x is plotted in the first quadrant. The vertical lines x equals a and x equals b are indicated by dashed lines. Four rectangles are fitted between x equals a and and x equals b where the height of each rectangle is approximately the height of the function f of x." style="margin: 10px 25px 25px 25px"/></center></li><li value="2"><p>
Add up the areas of all the rectangles. </p></li><li value="3"><p>
Take the limit as the rectangles become infinitesimally thin. </p></li></ol><p>
The main idea is that as the rectangles become thinner and thinner, the difference between the area covered by the rectangles and the area under the curve becomes smaller and smaller. And in the limit when the rectangles are infinitesimally small, these become exactly equal. </p>
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You can use the Mathlet below to investigate how the Riemann sum approaches the exact area under the curve by adjusting [mathjaxinline]n[/mathjaxinline]. </p><iframe src="https://mathlets1801.surge.sh/riemannSums.html" width="1100 px" height="630 px" style="border:0px; display: block;"/>
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Approximating the area under increasing functions 1
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<img alt="An increasing function y equals f of x is plotted in the first quadrant. The vertical lines x equals a and x equals b are indicated by dashed lines. Four rectangles are fitted between x equals a and and x equals b, dividing the interval a comma b into four subintervals. There are two situations that determine the height of the rectangles. In the first situation, the height of each rectangle is taken to be the height of the funcion at the left endpoint of the subintervals. In the second situation, the height of each rectangle is taken to be the height of the funciton at the right endpoint of the subintervals." src="/assets/courseware/v1/60338bf854e0b496570d08cbbd51a130/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_riemann2.svg" style="margin: 10px 25px 25px 25px" width="250px"/>
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Let [mathjaxinline]f(x)[/mathjaxinline] be an increasing function. We approximate the area under the curve in two ways. <br/>In the first way, we use rectangles whose heights are the function values at the left endpoints; in the second, we use right endpoints. </p>
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Determine the correct relationship between the exact area and the given approximation of the area. (I.e. is the approximation an over or under estimate.) <br/><p style="display:inline">[mathjaxinline]\displaystyle \int _ a^ b f(x) dx\, \,[/mathjaxinline] </p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input inline">
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<option value="[mathjaxinline]\geq[/mathjaxinline]"> [mathjaxinline]\geq[/mathjaxinline]</option>
<option value="[mathjaxinline]\leq[/mathjaxinline]"> [mathjaxinline]\leq[/mathjaxinline]</option>
<option value="can be either [mathjaxinline]\geq[/mathjaxinline] or [mathjaxinline]\leq[/mathjaxinline]"> can be either [mathjaxinline]\geq[/mathjaxinline] or [mathjaxinline]\leq[/mathjaxinline]</option>
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</div></div><p style="display:inline">Total area of all rectangles with top LEFT corners touching the graph of [mathjaxinline]f(x)[/mathjaxinline].</p><br/></p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _ a^ b f(x) dx\, \,[/mathjaxinline] </p>
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<p style="display:inline">Total area of all rectangles with top RIGHT corners touching the graph of [mathjaxinline]f(x)[/mathjaxinline].</p>
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Approximating the area under increasing functions 2
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<img alt="An increasing function y equals f of x is plotted in the first quadrant. The vertical lines x equals a and x equals b are indicated by dashed lines. Four rectangles are fitted between x equals a and and x equals b, dividing the interval a comma b into four subintervals. The height of each rectangle is taken to be the height of the funcion at the midpoint of the subintervals." src="/assets/courseware/v1/f90ada5e0a5d7953449e8a74bf1ebbfe/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_riemann3.svg" style="margin: 10px 25px 25px 25px" width="250px"/>
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As above, let [mathjaxinline]f(x)[/mathjaxinline] be an increasing function. This time, we will approximate the area under the graph using rectangles whose heights are the function value at the midtpoints. Determine the correct relationship between the exact area and the given the approximation of the area. (I.e. is the approximation an over or under estimate.) <br/><p style="display:inline">[mathjaxinline]\displaystyle \int _ a^ b f(x) dx\, \,[/mathjaxinline] </p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input inline">
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</div></div><p style="display:inline">Total area of all rectangles with MID-POINT of the top edges intersecting the graph of [mathjaxinline]f(x)[/mathjaxinline].</p><br/></p>
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The dimensions of the rectangles
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Our goal is to evaluate [mathjaxinline]\displaystyle \int _0^1 e^ x \, dx[/mathjaxinline] by the end of this section. Let us get started by approximating the area under [mathjaxinline]y=e^ x[/mathjaxinline] between [mathjaxinline]0[/mathjaxinline] and [mathjaxinline]1[/mathjaxinline] by the total area of the [mathjaxinline]n[/mathjaxinline] rectangles. We will choose the height of each rectangle is to be [mathjaxinline]e^ x[/mathjaxinline] evaluated at the LEFT endpoint of its base. <br/></p>
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<img alt="The function y equals e to the x is plotted in the first quadrant. The function intersects the y axis at the height 1. The vertical line x equals 1 is indicated. Thin rectangles divide the region beneath the function into subintervals. The height of each rectangle is taken to be the height of the function at the left endpoint of the subintervals. The total number of rectangles is n where rectangle 1 is the leftmost rectangle and rectangle n is the rightmost rectangle. Rectangle 3 is shaded." src="/assets/courseware/v1/b7757bf4224c13e0711c985d855cfced/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_rectangle3underexpbig.svg" style="margin: 10px 25px 25px 25px" width="350px"/>
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Find the dimensions and area of shaded rectangle 3, which is shaded in the figure above.<br/>(Your answer will be in terms of [mathjaxinline]n[/mathjaxinline].)<br/></p>
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<p style="display:inline">[mathjaxinline]\text {Base of rectangle 3} \, =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\text {Height of rectangle 3}\, =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\text {Area of rectangle 3}\, =\,[/mathjaxinline]</p>
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Total area of the rectangles
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<img alt="The function y equals e to the x is plotted in the first quadrant. The function intersects the y axis at the height 1. The vertical line x equals 1 is indicated. Thin rectangles divide the region beneath the function into subintervals. The height of each rectangle is taken to be the height of the function at the left endpoint of the subintervals. The total number of rectangles is n where rectangle 1 is the leftmost rectangle and rectangle n is the rightmost rectangle. All the rectangles are shaded." src="/assets/courseware/v1/a132e2bbb3de3afad53cfc4597f3292c/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_nrectanglesunderexpbig.svg" style="margin: 10px 25px 25px 25px" width="350px"/>
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<p>
Let us continue the approximation of [mathjaxinline]\displaystyle \int _0^1 e^ x \, dx[/mathjaxinline] by the total area of the [mathjaxinline]n[/mathjaxinline] rectangles. Recall that each rectangle has height [mathjaxinline]e^ x[/mathjaxinline] evaluated at the left endpoint of its base. <br/></p>
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What is the sum of the areas of the [mathjaxinline]n[/mathjaxinline] rectangles? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory1-tab8-problem2_2_1">
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<input type="radio" name="input_theory1-tab8-problem2_2_1" id="input_theory1-tab8-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory1-tab8-problem2_2_1-choice_1-label" for="input_theory1-tab8-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory1-tab8-problem2_2_1"> <text> [mathjaxinline]\displaystyle 1+e^{\frac{1}{n}}+e^{\frac{2}{n}}+e^{\frac{3}{n}}+\cdots +e^{\frac{n-1}{n}}[/mathjaxinline]</text>
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<p><b class="bfseries">Summation notation</b></p><p>
The [mathjaxinline]\sum[/mathjaxinline] notation is a compact way to denote a sum in which each term is obtained from a formula: <br/></p><table id="a0000000755" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \sum _{i=1}^ n a_ i = a_1+a_2+\cdots +a_{n-1}+a_{n}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]i[/mathjaxinline] indexes the terms, and [mathjaxinline]a_ i[/mathjaxinline] is a formula for the [mathjaxinline]i^{\text {th}}[/mathjaxinline] term of the sum. <br/>The notation [mathjaxinline]\displaystyle \sum _{i=1}^ n a_ i[/mathjaxinline] reads “the sum of [mathjaxinline]a_ i[/mathjaxinline] from [mathjaxinline]i=1[/mathjaxinline] to [mathjaxinline]i=n[/mathjaxinline]."<br/></p><p>
For example, if [mathjaxinline]a_ i=i^2[/mathjaxinline], that is, the formula for the term indexed by [mathjaxinline]i[/mathjaxinline] is [mathjaxinline]i^2[/mathjaxinline], then </p><table id="a0000000756" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \sum _{i=1}^ n i^2 = 1^2+2^2+3^2+\cdots +(n-1)^2+n^2[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This notation is a shorthand to write a potentially very long sum, but it does not simplify any calculations. </p>
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<p style="display:inline"> [mathjaxinline]\displaystyle \sum _{k=1}^{1001} (-1)^{k} =[/mathjaxinline]</p>
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Let [mathjaxinline]N\geq 4[/mathjaxinline] and [mathjaxinline]M\geq 0[/mathjaxinline]. <p style="display:inline"> [mathjaxinline]\displaystyle \sum _{i=4}^{N} 10 + \sum _{j=0}^{M} 6=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory1-tab9-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p style="display:inline"> \( n= \)</p>
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Pack the sum 2
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Find a [mathjaxinline]\sum[/mathjaxinline] notation expression for: <p style="display:inline">[mathjaxinline]\displaystyle 3-5+7-9+11-13:[/mathjaxinline]</p> </p>
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Find a [mathjaxinline]\sum[/mathjaxinline] notation expression for: <p style="display:inline">[mathjaxinline]\displaystyle \frac{1}{2k}+\frac{2}{3k}+\frac{3}{4k}+\cdots +\frac{k-1}{k^2}+\frac{1}{k+1}:[/mathjaxinline]</p> </p>
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Evaluate the sum
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Given the identity </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000764" style="table-layout:auto" width="100%">
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle 1+2+\cdots + n \, \, =\, \, \frac{n(n+1)}{2}[/mathjax]</td>
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Let [mathjaxinline]b\geq 0[/mathjaxinline].<br/></p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^{b} 1 \, dx =[/mathjaxinline]</p>
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<h3 class="hd hd-2">Summary of integral of powers</h3>
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<p><b class="bfseries">Summary of the definite integral of powers</b></p><p>
We have worked hard to evaluate our first non-trivial definite integral. Let us restate this result along with the definite integrals of lower powers of [mathjaxinline]x[/mathjaxinline] that we have evaluated in previous problems.<br/></p><table id="a0000000773" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000774"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \int _0^ b 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 \frac{b^3}{3}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.25)</td></tr><tr id="a0000000775"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int _0^ b 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 \int _0^ b x^1 \, 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 \frac{b^2}{2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.26)</td></tr><tr id="a0000000776"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int _0^ b 1 \, 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 \int _0^ b x^0 \, 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 b[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.27)</td></tr></table><p>
This suggests the following pattern </p><table id="a0000000777" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000778"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _0^ b x^ k \, 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 \frac{b^{k+1}}{k+1}\qquad \text {for any } k \neq -1,[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.28)</td></tr></table><p>
But how would we verify this pattern? <br/></p><p>
In fact, how would we evaluate other non-trivial definite integrals? <br/></p>
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<p><b class="bfseries">Riemann sums</b></p><p>
Let us summarize in precise terms the steps for evaluating [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline] using a Riemann Sum.<br/></p><ol class="enumerate"><li value="1"><p>
Divide [mathjaxinline][a,b][/mathjaxinline] into [mathjaxinline]n[/mathjaxinline] equal subintervals. </p><center><img src="/assets/courseware/v1/4bd8e2a2306320ad4f8695b1ba73c9c4/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_nintervals2.svg" width="350px" alt=" " style="margin: 10px 25px 25px 25px"/></center><p>
Then each interval is of length </p><table id="a0000000779" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \Delta x= \frac{b-a}{n}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Let the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval be the <b class="bfseries">base</b> of [mathjaxinline]i^{\text {th}}[/mathjaxinline] rectangle. </p></li><li value="2"><p>
Choose a point [mathjaxinline]c_ i[/mathjaxinline] within the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval.Choose [mathjaxinline]f(c_ i)[/mathjaxinline] be the <b class="bfseries">height</b> of the [mathjaxinline]i^{\text {th}}[/mathjaxinline] rectangle.<br/></p><center><img src="/assets/courseware/v1/d14e81fca4f42c812f7334c2c9fe3594/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_ci2.svg" width="350px" alt=" " style="margin: 10px 25px 25px 25px"/></center></li><li value="3"><p>
Add up the areas of the [mathjaxinline]n[/mathjaxinline] rectangles. The total area of [mathjaxinline]n[/mathjaxinline] rectangles is: </p><table id="a0000000780" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\underbrace{f(c_1)}_\text {height}\, \underbrace{\Delta x}_\text {base} +\underbrace{f(c_2)}_\text {height} \, \underbrace{\Delta x}_\text {base}+\cdots + \underbrace{f(c_ n)}_\text {height} \underbrace{\Delta x}_\text {base} \ =\ \sum _{i=1}^{n} f(c_ i) \Delta x[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li><li value="4"><p>
Take the limit as the rectangles become infinitesimally thin, ([mathjaxinline]\Delta x \rightarrow 0[/mathjaxinline], or equivalent [mathjaxinline]n\rightarrow \infty[/mathjaxinline]). This limit is the actual area under the curve between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p><table id="a0000000781" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\lim _{n\rightarrow \infty }\, \sum _{i=1}^{n} \, f(c_ i) \Delta x\ =\ \int _ a^ b f(x) \, dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li></ol><p>
The sum of the areas of the [mathjaxinline]n[/mathjaxinline] rectangles, [mathjaxinline]\sum _{i=1}^{n} f(c_ i) \Delta x[/mathjaxinline], is called a <span style="color:#27408C"><b class="bf">Riemann Sum</b></span>. If we pick [mathjaxinline]c_ i[/mathjaxinline] to be the left endpoint of the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval, the Riemann sum is called a <span style="color:#27408C"><b class="bf">left Riemann Sum</b></span>. Similarly, if [mathjaxinline]c_ i[/mathjaxinline] is the right endpoint of the [mathjaxinline]i^{\text {th}}[/mathjaxinline] interval, the Riemann sum is called a <span style="color:#27408C"><b class="bf">right Riemann sum</b></span>. <br/></p><p>
However, in the limit [mathjaxinline]n\rightarrow \infty[/mathjaxinline] (so that [mathjaxinline]\Delta x\rightarrow 0[/mathjaxinline]), this distinction is no longer needed. The limit of any Riemann Sum, no matter what the [mathjaxinline]c_ i[/mathjaxinline]'s within the subinterval are, is equal to the exact area under the curve. <br/></p>
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Computation of Riemann sums 1
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Estimate [mathjaxinline]\displaystyle \int _{0}^1 x^3\, \text {d}x[/mathjaxinline] using a Riemann sum. </p>
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<p style="display:inline">Right Riemann Sum with 4 equal subintervals:</p>
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Computation of Riemann sums 2
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<p style="display:inline">Use the distribution law and the answer to the previous problem to obtain an approximation of [mathjaxinline]\displaystyle \int _{0}^1 37 x^3\, \text {d}x[/mathjaxinline] using:<br/>Left Riemann Sum with 4 equal subintervals: </p>
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Definite integrals of constant multiples
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<p>
Let [mathjaxinline]k&gt;0[/mathjaxinline] be a constant.<br/>In all of the choices below, the Riemann sums are over an interval [mathjaxinline][a,b][/mathjaxinline] of length [mathjaxinline]1[/mathjaxinline] which is divided into equal subintervals with the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval labeled [mathjaxinline][x_{i-1},x_ i][/mathjaxinline]. Hence, the left and right Riemann sums with this labeling are as follows.<br/></p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000790" style="table-layout:auto" width="100%">
<tr id="a0000000791">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \text {Left Riemann sum of }\, \, n\, \, \text {subintervals :}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sum _{i=1}^{n} \frac{f(x_{i-1})}{n}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.34)</td>
</tr>
<tr id="a0000000792">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Right Riemann sum of }\, \, n\, \, \text {subintervals :}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sum _{i=1}^{n} \frac{f(x_{i})}{n}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.35)</td>
</tr>
</table>
<p>
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_theory1-tab13-problem3_2_1">
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<input type="checkbox" name="input_theory1-tab13-problem3_2_1[]" id="input_theory1-tab13-problem3_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="theory1-tab13-problem3_2_1-choice_0-label" for="input_theory1-tab13-problem3_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem3_2_1"> <text>[mathjaxinline]\displaystyle \sum _{i=1}^{100} 3\frac{f(x_{i-1})}{100} \, =\, 3 \sum _{i=1}^{100} \frac{f(x_{i-1})}{100}[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_theory1-tab13-problem3_2_1[]" id="input_theory1-tab13-problem3_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="theory1-tab13-problem3_2_1-choice_1-label" for="input_theory1-tab13-problem3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem3_2_1"> <text>[mathjaxinline]\displaystyle \sum _{i=1}^{100} k\frac{f(x_{i-1})}{100} \, =\, k \sum _{i=1}^{100} \frac{f(x_{i-1})}{100}[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_theory1-tab13-problem3_2_1[]" id="input_theory1-tab13-problem3_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="theory1-tab13-problem3_2_1-choice_2-label" for="input_theory1-tab13-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem3_2_1"> <text>[mathjaxinline]\displaystyle \sum _{i=1}^{100} k\frac{f(x_{i-1})}{100} \, =\, k \sum _{i=1}^{100} \frac{f(x_{i})}{100}[/mathjaxinline]</text>
</label>
</div>
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<input type="checkbox" name="input_theory1-tab13-problem3_2_1[]" id="input_theory1-tab13-problem3_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory1-tab13-problem3_2_1-choice_3-label" for="input_theory1-tab13-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem3_2_1"> <text>[mathjaxinline]\displaystyle \sum _{i=1}^{100} k\frac{f(x_{i-1})}{100} \, =\, k \sum _{i=1}^{150} \frac{f(x_{i-1})}{150}[/mathjaxinline]</text>
</label>
</div>
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<input type="checkbox" name="input_theory1-tab13-problem3_2_1[]" id="input_theory1-tab13-problem3_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="theory1-tab13-problem3_2_1-choice_4-label" for="input_theory1-tab13-problem3_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem3_2_1"> <text>[mathjaxinline]\displaystyle \sum _{i=1}^{n} x_{i-1}^2 \frac{f(x_{i-1})}{n} \, =\, x_{i-1}^2\cdot \sum _{i=1}^{n} \frac{f(x_{i-1})}{n}[/mathjaxinline]</text>
</label>
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<div class="field">
<input type="checkbox" name="input_theory1-tab13-problem3_2_1[]" id="input_theory1-tab13-problem3_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="theory1-tab13-problem3_2_1-choice_5-label" for="input_theory1-tab13-problem3_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem3_2_1"> <text>[mathjaxinline]\displaystyle \sum _{i=1}^{n} x_{i-1}^2 \frac{f(x_{i-1})}{n^2} \, =\, \left(\sum _{i=1}^{n} \frac{\left(x_{i-1}\right)^2}{n}\right) \cdot \left(\sum _{i=1}^{n} \frac{f(x_{i-1})}{n}\right)[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_theory1-tab13-problem3_2_1[]" id="input_theory1-tab13-problem3_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="theory1-tab13-problem3_2_1-choice_6-label" for="input_theory1-tab13-problem3_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem3_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty }\sum _{i=1}^{n} k\frac{f(x_{i-1})}{n} \, =\, k \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{f(x_{i-1})}{n}[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_theory1-tab13-problem3_2_1[]" id="input_theory1-tab13-problem3_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="theory1-tab13-problem3_2_1-choice_7-label" for="input_theory1-tab13-problem3_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem3_2_1"> <text>[mathjaxinline]\displaystyle \int _ a^ b kf(x)\, dx\, =\, k\cdot \int _ a^ b f(x) \, dx\,[/mathjaxinline]</text>
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Computation of Riemann sums 3
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<p style="display:inline">Estimate [mathjaxinline]\displaystyle \int _{0}^1 \left( \sin (\pi x)+37x^3\right)\, \text {d}x[/mathjaxinline] using:<br/>Left Riemann Sum with 4 equal subintervals: </p>
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Definite integrals of sums
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As above, in all of the choices below, the Riemann sums are over an interval [mathjaxinline][a,b][/mathjaxinline] of length [mathjaxinline]1[/mathjaxinline] which is divided into equal subintervals with the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval labeled [mathjaxinline][x_{i-1},x_ i][/mathjaxinline]. Hence, the left and right Riemann sums with this labeling are as follows.<br/></p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000818" style="table-layout:auto" width="100%">
<tr id="a0000000819">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \text {Left Riemann sum of }\, \, n\, \, \text {subintervals :}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sum _{i=1}^{n} \frac{f(x_{i-1})}{n}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.51)</td>
</tr>
<tr id="a0000000820">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Right Riemann sum of }\, \, n\, \, \text {subintervals :}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sum _{i=1}^{n} \frac{f(x_{i})}{n}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.52)</td>
</tr>
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Assume that all limits in this problem exist.<br/>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_theory1-tab13-problem5_2_1">
<fieldset aria-describedby="status_theory1-tab13-problem5_2_1">
<div class="field">
<input type="checkbox" name="input_theory1-tab13-problem5_2_1[]" id="input_theory1-tab13-problem5_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="theory1-tab13-problem5_2_1-choice_0-label" for="input_theory1-tab13-problem5_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem5_2_1"> <text>[mathjaxinline]\displaystyle \sum _{i=1}^{n} \frac{f(x_{i-1})+ g(x_{i-1} )}{n} \, =\, \sum _{i=1}^{n} \frac{f(x_{i-1})}{n}+\sum _{i=1}^{n} \frac{g(x_{i-1})}{n}[/mathjaxinline]</text>
</label>
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<div class="field">
<input type="checkbox" name="input_theory1-tab13-problem5_2_1[]" id="input_theory1-tab13-problem5_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="theory1-tab13-problem5_2_1-choice_1-label" for="input_theory1-tab13-problem5_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem5_2_1"> <text>[mathjaxinline]\displaystyle \sum _{i=1}^{n} \frac{f(x_{i-1})+ g(x_{i-1} )}{n}\, =\, \sum _{i=1}^{n} \frac{f(x_{i-1})}{n}+ \sum _{i=1}^{n} \frac{g(x_{i})}{n}[/mathjaxinline]</text>
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<div class="field">
<input type="checkbox" name="input_theory1-tab13-problem5_2_1[]" id="input_theory1-tab13-problem5_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="theory1-tab13-problem5_2_1-choice_2-label" for="input_theory1-tab13-problem5_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem5_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{f(x_{i-1})+ g(x_{i-1} )}{n}\, =\, \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{f(x_{i-1})}{n} + \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{g(x_{i-1})}{n}[/mathjaxinline]</text>
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<div class="field">
<input type="checkbox" name="input_theory1-tab13-problem5_2_1[]" id="input_theory1-tab13-problem5_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory1-tab13-problem5_2_1-choice_3-label" for="input_theory1-tab13-problem5_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem5_2_1"> <text>[mathjaxinline]\displaystyle \int _ a^ b \left(f(x)+g(x)\right)\, dx\, =\, \int _ a^ b f(x)\, dx + \int _ a^ b g(x) \, dx[/mathjaxinline]</text>
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<input type="checkbox" name="input_theory1-tab13-problem5_2_1[]" id="input_theory1-tab13-problem5_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="theory1-tab13-problem5_2_1-choice_4-label" for="input_theory1-tab13-problem5_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory1-tab13-problem5_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{f(x_{i-1})+ g(x_{i-1} )}{n}\, =\, \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{f(x_{i-1})}{n}+ \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{g(x_{i})}{n}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">14. Recognizing Riemann Sums</h2>
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Limit of sums can be very hard to evaluate. Recognizing a limit of sums as the limit of a Riemann sum allows us to evaluate the limit as the integral. <br/></p><p><b class="bf">Example problem:</b> Consider the right Riemann sum </p><table id="a0000000832" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\sum _{i=1}^{n} \frac{2}{n} \left(-1+\frac{2i}{n}\right)^3[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Express the limit </p><table id="a0000000833" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{2}{n} \left(-1+\frac{2i}{n}\right)^3[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
as a definite integral. </p><p><b class="bfseries"><span style="color:#FF7F00">Solution:</span></b> Let's evaluate the expressions inside the sum. </p><table id="a0000000834" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\sum _{i=1}^{n} \frac{2}{n} \left(-1+\frac{2i}{n}\right)^3[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
We want to write this sum in terms of a multiple [mathjaxinline]{\color{orange}{\Delta x}}[/mathjaxinline] that is independent of the index [mathjaxinline]i[/mathjaxinline] and a function [mathjaxinline]{\color{blue}{f(x)}}[/mathjaxinline] which should contain all of the terms involving the index [mathjaxinline]i[/mathjaxinline]. In our case </p><table id="a0000000835" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]{\color{orange}{\frac{2}{n}}} {\color{blue}{\left(-1+\frac{2i}{n}\right)^3}} = {\color{orange}{\Delta x}} {\color{blue}{ f(x)}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
At this stage what do we know? Since [mathjaxinline]\Delta x = 2/n[/mathjaxinline], the width of each rectangle is [mathjaxinline]2/n[/mathjaxinline] and we are summing over [mathjaxinline]n[/mathjaxinline] rectangles. Therefore the total length of the integral we are summing over is [mathjaxinline]n\cdot 2/n = 2[/mathjaxinline]. </p><p>
To determine the upper and lower limit of the definite integral, first we need to figure out what our function is. In this case it seems that </p><table id="a0000000836" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\left(-1+\frac{2i}{n}\right)^3 = f(-1+\frac{2i}{n}) = f(x) = x^3[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
so our function is that we are integrating is [mathjaxinline]f(x) = x^3[/mathjaxinline], and our expression for [mathjaxinline]x[/mathjaxinline] is [mathjaxinline]-1 +2i/n[/mathjaxinline]. In particular, when [mathjaxinline]i=1[/mathjaxinline], [mathjaxinline]x=-1+2/n[/mathjaxinline]. As [mathjaxinline]n[/mathjaxinline] tends to infinity, this becomes [mathjaxinline]x=-1[/mathjaxinline]. So the lower limit of the integral is [mathjaxinline]-1[/mathjaxinline]. When [mathjaxinline]i=n[/mathjaxinline], then [mathjaxinline]x=-1+2n/n = 1[/mathjaxinline]. Therefore the upper limit of the definite integral is [mathjaxinline]1[/mathjaxinline]. This confirms what we discovered earlier, the length of the integral we are integrating over is [mathjaxinline]1 - (-1) = 2[/mathjaxinline]. </p><p>
Therefore the limit of this sum represents the definite integral </p><table id="a0000000837" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{-1}^{1} x^3 \, dx \, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Recognizing Riemann Sums(*)
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[mathjaxinline]\displaystyle \int _1^2 \frac{1}{x}\, dx[/mathjaxinline]=?<br/><i class="itshape">Hint: Find an expression for [mathjaxinline]\Delta x[/mathjaxinline], then start by pulling out the term [mathjaxinline]\Delta x[/mathjaxinline] from each expression.</i> </p>
<p>
(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_theory1-tab14-problem1_2_1">
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<input type="checkbox" name="input_theory1-tab14-problem1_2_1[]" id="input_theory1-tab14-problem1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="theory1-tab14-problem1_2_1-choice_1-label" for="input_theory1-tab14-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory1-tab14-problem1_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=1}^ n \frac{1}{n+i}[/mathjaxinline]</text>
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<input type="checkbox" name="input_theory1-tab14-problem1_2_1[]" id="input_theory1-tab14-problem1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="theory1-tab14-problem1_2_1-choice_2-label" for="input_theory1-tab14-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory1-tab14-problem1_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=0}^{n-1} \frac{1}{n+i}[/mathjaxinline]</text>
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<input type="checkbox" name="input_theory1-tab14-problem1_2_1[]" id="input_theory1-tab14-problem1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory1-tab14-problem1_2_1-choice_3-label" for="input_theory1-tab14-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory1-tab14-problem1_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{1}{n+i-1}[/mathjaxinline]</text>
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<input type="checkbox" name="input_theory1-tab14-problem1_2_1[]" id="input_theory1-tab14-problem1_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="theory1-tab14-problem1_2_1-choice_4-label" for="input_theory1-tab14-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory1-tab14-problem1_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{n}{n+i}[/mathjaxinline]</text>
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<input type="checkbox" name="input_theory1-tab14-problem1_2_1[]" id="input_theory1-tab14-problem1_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="theory1-tab14-problem1_2_1-choice_5-label" for="input_theory1-tab14-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_theory1-tab14-problem1_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \frac{n}{i}[/mathjaxinline]</text>
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<input type="checkbox" name="input_theory1-tab14-problem1_2_1[]" id="input_theory1-tab14-problem1_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="theory1-tab14-problem1_2_1-choice_6-label" for="input_theory1-tab14-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_theory1-tab14-problem1_2_1"> <text>[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=0}^{n-1} \frac{2}{2n+1+2i}[/mathjaxinline]</text>
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An infinite sum as an integral(*)
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Rewrite the following limit as a definite integral. </p>
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[mathjaxinline]\displaystyle \lim _{k \rightarrow \infty } \sum _{n=1}^{k} \frac{1}{k} \frac{n^2+3nk+9k^2 \sin (n/k)}{k^2}=[/mathjaxinline] </p>
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(Enter the upper limit in the answer box above the [mathjaxinline]\int[/mathjaxinline], the lower limit of integration in the answer box below the [mathjaxinline]\int[/mathjaxinline], and the integrand to the right of the [mathjaxinline]\int[/mathjaxinline]. The [mathjaxinline]dx[/mathjaxinline] is provided for you. All 3 answer boxes are graded together by numerically integrating your integral.) </p>
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<h2 class="hd hd-2 unit-title">15. Compute the definite integral of exp</h2>
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Riemann sum of the exponential in summation notation(*)
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In these three problems, we will evaluate our second non-trivial definite integral, [mathjaxinline]\displaystyle \int _0^1 e^ x \, dx[/mathjaxinline], as the limit of a Riemann Sum. </p>
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In a previous problem, we have found the the total area of the [mathjaxinline]n[/mathjaxinline] rectangles used in the left Riemann sum, shaded in the figure below. </p>
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Write the total area of the [mathjaxinline]n[/mathjaxinline] rectangles in the left Riemann sum in the [mathjaxinline]\displaystyle \sum[/mathjaxinline] notation: </p>
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<p> \( \displaystyle \huge{ \sum }\)</p>
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Evaluating the sum(*)
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As above, our goal is to evaluate [mathjaxinline]\displaystyle \int _0^1 e^ x \, dx[/mathjaxinline] as the limit of its Riemann Sum.<br/></p>
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Find a closed form expression (without a sum) for the total area of the [mathjaxinline]n[/mathjaxinline] rectangles (which you have written in the [mathjaxinline]\sum[/mathjaxinline] notation in the previous problem,) by using the formula below for the first [mathjaxinline]n^{\text {th}}[/mathjaxinline] terms of a geometric series. </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \sum _{k=0}^{n-1} r^ k\, =\, \frac{1-r^{n}}{1-r}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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(Enter your answer in terms of [mathjaxinline]n[/mathjaxinline].)<br/></p>
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<p style="display:inline">Total area of the [mathjaxinline]n[/mathjaxinline] rectangles [mathjaxinline]=[/mathjaxinline]</p>
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Taking the limit(*)
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As the final step, evaluate [mathjaxinline]\displaystyle \int _0^1 e^ x \, dx[/mathjaxinline]. In other words, take the limit as [mathjaxinline]n\rightarrow \infty[/mathjaxinline] of the expression you found above for the total area of the [mathjaxinline]n[/mathjaxinline] rectangles in the left Riemann Sum. Enter your answer below </p>
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Units of the Riemann sum
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<p>
Let us explore the units of a Riemann sum.</p>
<p>
Suppose [mathjaxinline]f(t)[/mathjaxinline] is the velocity of your car, measured in [mathjaxinline]\text {km/hour}[/mathjaxinline], at time [mathjaxinline]t[/mathjaxinline], measured in [mathjaxinline]\text {hours}[/mathjaxinline]. </p>
<p>
Divide the total time of the journey in [mathjaxinline]n[/mathjaxinline] time intervals each of length [mathjaxinline]\Delta t[/mathjaxinline], and let [mathjaxinline]c_ i[/mathjaxinline] be a moment within the [mathjaxinline]i^{\text {th}}[/mathjaxinline] time interval. That is, as components in a Riemann sum of [mathjaxinline]f(t)[/mathjaxinline], [mathjaxinline]\Delta t[/mathjaxinline] is the base of any rectangle and [mathjaxinline]c_ i[/mathjaxinline] is any point on the base of the [mathjaxinline]i^\text {th}[/mathjaxinline] rectangle.<br/>Determine the units of the quantities below<br/></p>
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<tr>
<td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \Delta t \cdot f(c_1)[/mathjaxinline]?</td>
<td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sum _{i=1}^{n} \Delta t \cdot f(c_ i)[/mathjaxinline]?</td>
<td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \lim _{n\rightarrow \infty } \sum _{i=1}^{n} \Delta t \cdot f(c_ i)[/mathjaxinline]?</td>
<td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \int _{a}^{b} f(t) \, dt[/mathjaxinline] </td>
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<text> [mathjaxinline]\text {hour}[/mathjaxinline]</text>
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<text> [mathjaxinline]\text {km}[/mathjaxinline]</text>
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<input type="radio" name="input_theory1-tab16-problem1_2_1" id="input_theory1-tab16-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory1-tab16-problem1_2_1-choice_3-label" for="input_theory1-tab16-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_2_1">
<text> [mathjaxinline]\text {km}/\text {hour}[/mathjaxinline]</text>
</label>
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<text> [mathjaxinline]\text {km}^2/\text {hour}[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_theory1-tab16-problem1_2_1" id="input_theory1-tab16-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="theory1-tab16-problem1_2_1-choice_5-label" for="input_theory1-tab16-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_2_1">
<text> [mathjaxinline]\text {km}/\text {hour}^2[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_theory1-tab16-problem1_2_1" id="input_theory1-tab16-problem1_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="theory1-tab16-problem1_2_1-choice_6-label" for="input_theory1-tab16-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_2_1">
<text> [mathjaxinline]\text {unitless}[/mathjaxinline]</text>
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<text> [mathjaxinline]\text {hr}[/mathjaxinline]</text>
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<input type="radio" name="input_theory1-tab16-problem1_3_1" id="input_theory1-tab16-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory1-tab16-problem1_3_1-choice_2-label" for="input_theory1-tab16-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_3_1">
<text> [mathjaxinline]\text {km}[/mathjaxinline]</text>
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<text> [mathjaxinline]\text {km}/\text {hr}[/mathjaxinline]</text>
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<text> [mathjaxinline]\text {km}^ n/\text {hour}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_3_1" id="input_theory1-tab16-problem1_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="theory1-tab16-problem1_3_1-choice_5-label" for="input_theory1-tab16-problem1_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_3_1">
<text> [mathjaxinline]\text {km}/\text {hour}^ n[/mathjaxinline]</text>
</label>
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<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_3_1" id="input_theory1-tab16-problem1_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="theory1-tab16-problem1_3_1-choice_6-label" for="input_theory1-tab16-problem1_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_3_1">
<text> [mathjaxinline]\text {unitless}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_theory1-tab16-problem1_3_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<input type="radio" name="input_theory1-tab16-problem1_4_1" id="input_theory1-tab16-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory1-tab16-problem1_4_1-choice_1-label" for="input_theory1-tab16-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_4_1">
<text> [mathjaxinline]\text {hour}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_4_1" id="input_theory1-tab16-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory1-tab16-problem1_4_1-choice_2-label" for="input_theory1-tab16-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_4_1">
<text> [mathjaxinline]\text {km}[/mathjaxinline]</text>
</label>
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<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_4_1" id="input_theory1-tab16-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory1-tab16-problem1_4_1-choice_3-label" for="input_theory1-tab16-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_4_1">
<text> [mathjaxinline]\text {km}/\text {hour}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_4_1" id="input_theory1-tab16-problem1_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="theory1-tab16-problem1_4_1-choice_4-label" for="input_theory1-tab16-problem1_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_4_1">
<text> [mathjaxinline]\text {km}^2/\text {hour}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_4_1" id="input_theory1-tab16-problem1_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="theory1-tab16-problem1_4_1-choice_5-label" for="input_theory1-tab16-problem1_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_4_1">
<text> [mathjaxinline]\text {km}/\text {hour}^2[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_4_1" id="input_theory1-tab16-problem1_4_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="theory1-tab16-problem1_4_1-choice_6-label" for="input_theory1-tab16-problem1_4_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_4_1">
<text> [mathjaxinline]\text {unitless}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_theory1-tab16-problem1_4_1"/>
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<span class="status unanswered" id="status_theory1-tab16-problem1_4_1" data-tooltip="Not yet answered.">
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory1-tab16-problem1_5_1">
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<input type="radio" name="input_theory1-tab16-problem1_5_1" id="input_theory1-tab16-problem1_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory1-tab16-problem1_5_1-choice_1-label" for="input_theory1-tab16-problem1_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_5_1">
<text> [mathjaxinline]\text {hour}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_5_1" id="input_theory1-tab16-problem1_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory1-tab16-problem1_5_1-choice_2-label" for="input_theory1-tab16-problem1_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_5_1">
<text> [mathjaxinline]\text {km}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_5_1" id="input_theory1-tab16-problem1_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory1-tab16-problem1_5_1-choice_3-label" for="input_theory1-tab16-problem1_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_5_1">
<text> [mathjaxinline]\text {km}/\text {hour}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_5_1" id="input_theory1-tab16-problem1_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="theory1-tab16-problem1_5_1-choice_4-label" for="input_theory1-tab16-problem1_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_5_1">
<text> [mathjaxinline]\text {km}^2/\text {hour}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_5_1" id="input_theory1-tab16-problem1_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="theory1-tab16-problem1_5_1-choice_5-label" for="input_theory1-tab16-problem1_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_5_1">
<text> [mathjaxinline]\text {km}/\text {hour}^2[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem1_5_1" id="input_theory1-tab16-problem1_5_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="theory1-tab16-problem1_5_1-choice_6-label" for="input_theory1-tab16-problem1_5_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem1_5_1">
<text> [mathjaxinline]\text {unitless}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_theory1-tab16-problem1_5_1"/>
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<span class="status unanswered" id="status_theory1-tab16-problem1_5_1" data-tooltip="Not yet answered.">
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<p><b class="bfseries">Cumulative sums</b></p><p>
We have been defining definite integrals geometrically as the area under a curve. But [mathjaxinline]f(t)[/mathjaxinline] and [mathjaxinline]t[/mathjaxinline] can represent quantities other than length. </p><p>
For instance, in the previous problem, [mathjaxinline]t[/mathjaxinline] represents time, with units of [mathjaxinline]\text {hours}[/mathjaxinline], and [mathjaxinline]f(t)[/mathjaxinline] represents velocity, with units of [mathjaxinline]\text {km/hour}[/mathjaxinline]. So the units of the integral, [mathjaxinline]\displaystyle \int _ a^ b f(t)\, dt[/mathjaxinline], is [mathjaxinline]\text {km}[/mathjaxinline], which are the units of distance. But the integral [mathjaxinline]\displaystyle \int _{a}^{b} f(t)\, dt[/mathjaxinline] still represents the area under the curve, so the units of this area are [mathjaxinline]\text {km}[/mathjaxinline].<br/></p><p>
Most applications of the integral will involve seeing it as a cumulative sum. You will see numerous examples in Unit 3. </p>
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Interpretation of the sum
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory1-tab16-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, suppose [mathjaxinline]f(t)\geq 0[/mathjaxinline] is the velocity of your car at time [mathjaxinline]t[/mathjaxinline], your journey lasts from time [mathjaxinline]a[/mathjaxinline] to time [mathjaxinline]b[/mathjaxinline], and is divided into [mathjaxinline]n[/mathjaxinline] time intervals each of length [mathjaxinline]\Delta t[/mathjaxinline]. Let [mathjaxinline]c_ i[/mathjaxinline] be a moment within the [mathjaxinline]i^{\text {th}}[/mathjaxinline] time interval. <br/></p>
<p>
What does [mathjaxinline]f(c_ i) \cdot \Delta t[/mathjaxinline] approximate?<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory1-tab16-problem2_2_1">
<fieldset aria-describedby="status_theory1-tab16-problem2_2_1">
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_2_1" id="input_theory1-tab16-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory1-tab16-problem2_2_1-choice_1-label" for="input_theory1-tab16-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_2_1"> <text> average acceleration of the car from [mathjaxinline]t_{i-1}[/mathjaxinline] to [mathjaxinline]t_ i[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_2_1" id="input_theory1-tab16-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory1-tab16-problem2_2_1-choice_2-label" for="input_theory1-tab16-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_2_1"> <text> change in velocity of the car from [mathjaxinline]t_{i-1}[/mathjaxinline] to [mathjaxinline]t_ i[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_2_1" id="input_theory1-tab16-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory1-tab16-problem2_2_1-choice_3-label" for="input_theory1-tab16-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_2_1"> <text> change in time from [mathjaxinline]t_{i-1}[/mathjaxinline] to [mathjaxinline]t_ i[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_2_1" id="input_theory1-tab16-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="theory1-tab16-problem2_2_1-choice_4-label" for="input_theory1-tab16-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_2_1"> <text> area the car covered from time [mathjaxinline]t_{i-1}[/mathjaxinline] to [mathjaxinline]t_ i[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_2_1" id="input_theory1-tab16-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="theory1-tab16-problem2_2_1-choice_5-label" for="input_theory1-tab16-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_2_1"> <text> total distance traveled from time [mathjaxinline]t_{i-1}[/mathjaxinline] to [mathjaxinline]t_ i[/mathjaxinline]</text>
</label>
</div>
<span id="answer_theory1-tab16-problem2_2_1"/>
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<span class="status unanswered" id="status_theory1-tab16-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p>
Hence, what does [mathjaxinline]\displaystyle \sum _{i=1}^ n f(c_ i) \cdot \Delta t[/mathjaxinline], the sum of the quantity above, approximate?<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory1-tab16-problem2_3_1">
<fieldset aria-describedby="status_theory1-tab16-problem2_3_1">
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_3_1" id="input_theory1-tab16-problem2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory1-tab16-problem2_3_1-choice_1-label" for="input_theory1-tab16-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_3_1"> <text> average acceleration of the car from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]b[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_3_1" id="input_theory1-tab16-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory1-tab16-problem2_3_1-choice_2-label" for="input_theory1-tab16-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_3_1"> <text> total change in velocity of the car from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]b[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_3_1" id="input_theory1-tab16-problem2_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory1-tab16-problem2_3_1-choice_3-label" for="input_theory1-tab16-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_3_1"> <text> total change in time from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]b[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_3_1" id="input_theory1-tab16-problem2_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="theory1-tab16-problem2_3_1-choice_4-label" for="input_theory1-tab16-problem2_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_3_1"> <text> total area that the car covered from time [mathjaxinline]t_{i-1}[/mathjaxinline] to [mathjaxinline]t_ i[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory1-tab16-problem2_3_1" id="input_theory1-tab16-problem2_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="theory1-tab16-problem2_3_1-choice_5-label" for="input_theory1-tab16-problem2_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_theory1-tab16-problem2_3_1"> <text> total distance traveled from time [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]b[/mathjaxinline]</text>
</label>
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<span id="answer_theory1-tab16-problem2_3_1"/>
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<p><b class="bfseries"><span style="color:#FF7F00">Note on video:</span></b> In the chemical diffusion example, you do not have the tools at this point to compute this integral. In the next lecture, when we learn the Fundamental Theorem of Calculus, you will be able to apply the hint Christine gave to compute this integral. </p>
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<h2 class="hd hd-2 unit-title">17. Summary</h2>
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<p><b class="bfseries">Geometric definition of the definite integral</b></p><p>
The <span style="color:#99182C"><b class="bf">definite integral of [mathjaxinline]f\,[/mathjaxinline] from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]b[/mathjaxinline]</b></span>, denoted by </p><table id="a0000000881" 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,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
is the area of the region above the [mathjaxinline]x[/mathjaxinline]-axis, below the curve [mathjaxinline]y=f(x)[/mathjaxinline], and in between the two vertical lines [mathjaxinline]x=a[/mathjaxinline] and [mathjaxinline]x=b[/mathjaxinline], as shown shaded in the figure below. </p><center><img src="/assets/courseware/v1/b4e1e194803ea64ed5decb84891c8a57/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_riemann4.svg" width="250px" alt="A function y equals f of x is plotted in the first quadrant. The points a and b are indicated on the x axis. The region beneath the function and between the vertical lines x equals a and x equals b is shaded." style="margin: 10px 25px 25px 25px"/></center><p>
The [mathjaxinline]x[/mathjaxinline]-values [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] are called the <span style="color:#99182C"><b class="bf">lower</b></span> and <span style="color:#99182C"><b class="bf">upper limits of the integral</b></span>. (This is a different sense of the word “limit" from when we take the limit of a function.) <br/></p><p>
The only difference between the notation for definite and indefinite integrals is that definite integrals have limits but indefinite integrals do not.<br/></p><p><b class="bfseries">Summation notation</b></p><p>
The [mathjaxinline]\sum[/mathjaxinline] notation in a compact way to denote a sum in which each term is obtained from a formula: <br/></p><table id="a0000000882" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \sum _{i=1}^ n a_ i = a_1+a_2+\cdots +a_{n-1}+a_{n}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]i[/mathjaxinline] indexes the terms, and [mathjaxinline]a_ i[/mathjaxinline] is a formula for the [mathjaxinline]i^{\text {th}}[/mathjaxinline] term of the sum. <br/>The notation [mathjaxinline]\displaystyle \sum _{i=1}^ n a_ i[/mathjaxinline] reads “the sum of [mathjaxinline]a_ i[/mathjaxinline] from [mathjaxinline]i=1[/mathjaxinline] to [mathjaxinline]i=n[/mathjaxinline]."<br/></p><p>
For example, if [mathjaxinline]a_ i=i^2[/mathjaxinline], that is, the formula for the term indexed by [mathjaxinline]i[/mathjaxinline] is [mathjaxinline]i^2[/mathjaxinline], then </p><table id="a0000000883" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \sum _{i=1}^ n i^2 = 1^2+2^2+3^2+\cdots +(n-1)^2+n^2[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Riemann sums</b></p><p>
Let us summarize in precise terms the steps for evaluating [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline] using a Riemann Sum.<br/></p><ol class="enumerate"><li value="1"><p>
Divide [mathjaxinline][a,b][/mathjaxinline] into [mathjaxinline]n[/mathjaxinline] equal subintervals. </p><center><img src="/assets/courseware/v1/4bd8e2a2306320ad4f8695b1ba73c9c4/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_nintervals2.svg" width="350px" alt=" " style="margin: 10px 25px 25px 25px"/></center><p>
Then each interval is of length </p><table id="a0000000884" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \Delta x= \frac{b-a}{n}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Let the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval be the <b class="bfseries">base</b> of [mathjaxinline]i^{\text {th}}[/mathjaxinline] rectangle. </p></li><li value="2"><p>
Choose a point [mathjaxinline]c_ i[/mathjaxinline] within the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval.Choose [mathjaxinline]f(c_ i)[/mathjaxinline] be the <b class="bfseries">height</b> of the [mathjaxinline]i^{\text {th}}[/mathjaxinline] rectangle.<br/></p><center><img src="/assets/courseware/v1/d14e81fca4f42c812f7334c2c9fe3594/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_defint_ci2.svg" width="350px" alt=" " style="margin: 10px 25px 25px 25px"/></center></li><li value="3"><p>
Add up the areas of the [mathjaxinline]n[/mathjaxinline] rectangles. The total area of [mathjaxinline]n[/mathjaxinline] rectangles is: </p><table id="a0000000885" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\underbrace{f(c_1)}_\text {height}\, \underbrace{\Delta x}_\text {base} +\underbrace{f(c_2)}_\text {height} \, \underbrace{\Delta x}_\text {base}+\cdots + \underbrace{f(c_ n)}_\text {height} \underbrace{\Delta x}_\text {base} \ =\ \sum _{i=1}^{n} f(c_ i) \Delta x[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li><li value="4"><p>
Take the limit as the rectangles become infinitesimally thin, ([mathjaxinline]\Delta x \rightarrow 0[/mathjaxinline], or equivalent [mathjaxinline]n\rightarrow \infty[/mathjaxinline]). This limit is the actual area under the curve between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p><table id="a0000000886" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\lim _{n\rightarrow \infty }\, \sum _{i=1}^{n} \, f(c_ i) \Delta x\ =\ \int _ a^ b f(x) \, dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li></ol><p>
The sum of the areas of the [mathjaxinline]n[/mathjaxinline] rectangles, [mathjaxinline]\sum _{i=1}^{n} f(c_ i) \Delta x[/mathjaxinline], is called a <span style="color:#27408C"><b class="bf">Riemann Sum</b></span>. If we pick [mathjaxinline]c_ i[/mathjaxinline] to be the left endpoint of the [mathjaxinline]i^{\text {th}}[/mathjaxinline] subinterval, the Riemann sum is called a <span style="color:#27408C"><b class="bf">left Riemann Sum</b></span>. Similarly, if [mathjaxinline]c_ i[/mathjaxinline] is the right endpoint of the [mathjaxinline]i^{\text {th}}[/mathjaxinline] interval, the Riemann sum is called a <span style="color:#27408C"><b class="bf">right Riemann sum</b></span>. <br/></p><p>
However, in the limit [mathjaxinline]n\rightarrow \infty[/mathjaxinline] (so that [mathjaxinline]\Delta x\rightarrow 0[/mathjaxinline]), this distinction is no longer needed. The limit of any Riemann Sum, no matter what the [mathjaxinline]c_ i[/mathjaxinline]'s within the subinterval are, is equal to the exact area under the curve. <br/></p><p><b class="bfseries">Cumulative sums</b></p><p>
We have been defining the definite integral [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline] geometrically as the area under a curve. But [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]x[/mathjaxinline] can represent quantities other than length. </p><p>
For instance, if [mathjaxinline]x[/mathjaxinline] represents time, with units of [mathjaxinline]\text {hours}[/mathjaxinline], and [mathjaxinline]f(x)[/mathjaxinline] represents velocity, with units of [mathjaxinline]\text {km/hour}[/mathjaxinline]. So the units of the integral, [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline], is [mathjaxinline]\text {km}[/mathjaxinline], which are the units of distance. But the integral [mathjaxinline]\displaystyle \int _{a}^{b} f(x)\, dx[/mathjaxinline] still represents the area under the curve, so the units of this area are [mathjaxinline]\text {km}[/mathjaxinline].<br/></p><p>
Most applications of the integral will involve seeing it as a cumulative sum. You will see numerous examples in Unit 3. </p>
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