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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Average value</h3>
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<h2 class="hd hd-2 unit-title">2. Average value and weighted averages</h2>
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<p><h3>Objectives</h3></p><ul class="itemize"><li><p>
Find the <span style="color:#27408C"><b class="bf">average value</b></span> of a continuous function. </p></li><li><p>
Interpret the average value <span style="color:#27408C"><b class="bf">geometrically</b></span>. </p></li><li><p>
Use integrals to compute <span style="color:#27408C"><b class="bf">work</b></span>. </p></li><li><p>
Compute <span style="color:#27408C"><b class="bf">weighted averages</b></span> of continuous functions. </p></li><li><p>
Interpret <span style="color:#27408C"><b class="bf">center of mass</b></span> as the average value of position weighted by mass density. </p></li></ul><p><h3>Contents: 13 pages</h3></p><p>
11 videos (48 minutes at 1x speed) 16 problems </p>
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<h2 class="hd hd-2 unit-title">3. Discrete averages</h2>
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<p>
When we talk about the <span style="color:#27408C"><b class="bf">average</b></span> in this course, we mean the <span style="color:#27408C"><b class="bf">mean</b></span>, which is the sum of all values divided by the number of values. There are other numbers used to represent the typical value. For example the <span style="color:#27408C"><b class="bf">median</b></span>, which is the value in the middle of the ordered list of values, and the <span style="color:#27408C"><b class="bf">mode</b></span>, which is the value that occurs most frequently. We are only going to discuss the mean because it is the one that is directly connected to integrals. </p><p>
We will review discrete average, and show how the limit gives rise to average values of continuous functions. </p>
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On a calculus exam given at MIT:<br/>1 student scored 100,<br/>3 students scored 95,<br/>2 students scored 82,<br/>7 students scored 73,<br/>and 2 students scored 68. </p>
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What was the average exam score? </p>
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<h2 class="hd hd-2 unit-title">4. Continuous averages</h2>
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<p>
We are learning calculus; we want to extend the idea of discrete averages to find the average value of a continuous function. </p><p>
What do we mean by the average height of a continuous function? To help us understand what we mean by continuous averages and how to compute them, we'll start with two examples that are intuitive. </p>
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What is the average value (average height) of the constant function [mathjaxinline]\ f(x) = c[/mathjaxinline]? </p>
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<p>
What is the average value (average height) of the straight line [mathjaxinline]\ f(x) = x/2 + 1[/mathjaxinline] on the interval [mathjaxinline]0 \leq x \leq 5[/mathjaxinline]? </p>
<center>
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Derive the formula
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab4-problem3-problem-progress"></div>
<div class="problem">
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<p>
Using the geometric interpretation of the average value of a continuous function, write the formula for the average value on the interval [mathjaxinline]\left[ a, \, b \right][/mathjaxinline], which we denote by [mathjaxinline]\mathrm{ave}(f)[/mathjaxinline]. </p>
<span>
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<h2 class="hd hd-2 unit-title">5. Continuous averages</h2>
<div class="vert-mod">
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class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab5-video1/handler/publish_completion", "streams": "1.00:DRE8AxB3meg", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab5-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab5-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab5-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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<h2 class="hd hd-2 unit-title">6. Examples</h2>
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<p><b class="bfseries">Average value of [mathjaxinline] f(x)[/mathjaxinline]</b></p><p>
The average value of a function [mathjaxinline]\ f(x)[/mathjaxinline] on an interval [mathjaxinline]a \leq x \leq b[/mathjaxinline] is given by </p><table id="a0000001455" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{1}{b-a}\int _ a^ b f(x)\, dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Average value computation
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Compute the average value of the function [mathjaxinline]\ f(x) = 1- x^2[/mathjaxinline] over the interval [mathjaxinline]-1 \leq x \leq 1[/mathjaxinline]. </p>
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Average velocity
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Let [mathjaxinline]y(t)=-5t^2+2t+11[/mathjaxinline] meters be the height of a ball. Then the velocity is [mathjaxinline]v(t) = y'(t)[/mathjaxinline]. Find the average velocity [mathjaxinline]\mathrm{ave}(v)[/mathjaxinline] over the interval [mathjaxinline]0 \leq t \leq 2[/mathjaxinline] seconds. </p>
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<span class="trailing_text" id="trailing_text_applications2-tab6-problem2_2_1">meters per second</span>
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<h3 class="hd hd-2">Example video: average velocity</h3>
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data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab6-video1/handler/publish_completion", "streams": "1.00:PKkaq8G1gps", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab6-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab6-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab6-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
data-bumper-metadata='null'
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tabindex="-1"
>
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Average cross-sectional area (*)
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What is the average cross-sectional area [mathjaxinline]A[/mathjaxinline] of the solid obtained by revolving the region bounded by [mathjaxinline]x=2[/mathjaxinline], the [mathjaxinline]x[/mathjaxinline]-axis, and the curve [mathjaxinline]y=x^2[/mathjaxinline] about the [mathjaxinline]x[/mathjaxinline]-axis? (Cross-sections are taken perpendicular to the [mathjaxinline]x[/mathjaxinline]-axis.) </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle A=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">7. MVT and FTC</h2>
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data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab7-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab7-problem1/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="applications2-tab7-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab7-problem1-problem-progress" tabindex="-1">
MVT, FTC1, and average value
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab7-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Let [mathjaxinline]F[/mathjaxinline] be a function that is differentiable for [mathjaxinline]a \leq x \leq b[/mathjaxinline]. </p>
<p>
Which of the following are equivalent to [mathjaxinline]\displaystyle \frac{F(b) - F(a)}{b-a}[/mathjaxinline]? (Choose all that apply. Note that each column is graded separately.) </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_applications2-tab7-problem1_2_1">
<fieldset aria-describedby="status_applications2-tab7-problem1_2_1">
<div class="field">
<input type="checkbox" name="input_applications2-tab7-problem1_2_1[]" id="input_applications2-tab7-problem1_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="applications2-tab7-problem1_2_1-choice_0-label" for="input_applications2-tab7-problem1_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_applications2-tab7-problem1_2_1">
<text>Average rate of change of [mathjaxinline]F[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_applications2-tab7-problem1_2_1[]" id="input_applications2-tab7-problem1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="applications2-tab7-problem1_2_1-choice_1-label" for="input_applications2-tab7-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_applications2-tab7-problem1_2_1">
<text>The average value of [mathjaxinline]F[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_applications2-tab7-problem1_2_1[]" id="input_applications2-tab7-problem1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="applications2-tab7-problem1_2_1-choice_2-label" for="input_applications2-tab7-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_applications2-tab7-problem1_2_1">
<text>[mathjaxinline]\displaystyle \frac{1}{b-a}\int _ a^ b F \, dx[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_applications2-tab7-problem1_2_1[]" id="input_applications2-tab7-problem1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="applications2-tab7-problem1_2_1-choice_3-label" for="input_applications2-tab7-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_applications2-tab7-problem1_2_1">
<text>[mathjaxinline]F(c)[/mathjaxinline] for some [mathjaxinline]c[/mathjaxinline] in [mathjaxinline](a, \, b)[/mathjaxinline].</text>
</label>
</div>
<span id="answer_applications2-tab7-problem1_2_1"/>
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<span class="status unanswered" id="status_applications2-tab7-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</td>
<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_applications2-tab7-problem1_3_1">
<fieldset aria-describedby="status_applications2-tab7-problem1_3_1">
<div class="field">
<input type="checkbox" name="input_applications2-tab7-problem1_3_1[]" id="input_applications2-tab7-problem1_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="applications2-tab7-problem1_3_1-choice_0-label" for="input_applications2-tab7-problem1_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_applications2-tab7-problem1_3_1">
<text>Average rate of change of [mathjaxinline]F'[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_applications2-tab7-problem1_3_1[]" id="input_applications2-tab7-problem1_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="applications2-tab7-problem1_3_1-choice_1-label" for="input_applications2-tab7-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_applications2-tab7-problem1_3_1">
<text>The average value of [mathjaxinline]F'[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_applications2-tab7-problem1_3_1[]" id="input_applications2-tab7-problem1_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="applications2-tab7-problem1_3_1-choice_2-label" for="input_applications2-tab7-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_applications2-tab7-problem1_3_1">
<text>[mathjaxinline]\displaystyle \frac{1}{b-a} \int _ a^ b F' \, dx[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_applications2-tab7-problem1_3_1[]" id="input_applications2-tab7-problem1_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="applications2-tab7-problem1_3_1-choice_3-label" for="input_applications2-tab7-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_applications2-tab7-problem1_3_1">
<text>[mathjaxinline]F'(c)[/mathjaxinline] for some [mathjaxinline]c[/mathjaxinline] in [mathjaxinline](a, \, b)[/mathjaxinline].</text>
</label>
</div>
<span id="answer_applications2-tab7-problem1_3_1"/>
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<span class="status unanswered" id="status_applications2-tab7-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</td>
</tr>
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<p>
<div class="solution-span">
<span id="solution_applications2-tab7-problem1_solution_1"/>
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aria-labelledby="applications2-tab7-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab7-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab7-problem2/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="applications2-tab7-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab7-problem2-problem-progress" tabindex="-1">
Applying FTC2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications2-tab7-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
If the average value of [mathjaxinline]f(t)[/mathjaxinline] between [mathjaxinline]0[/mathjaxinline] and [mathjaxinline]x[/mathjaxinline] is given by the function [mathjaxinline]g(x)[/mathjaxinline]: </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000001464" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle g(x) = \frac{1}{x} \int _0^ x f(t)\, dt.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Write [mathjaxinline]f(x)[/mathjaxinline] in terms of [mathjaxinline]g(x)[/mathjaxinline] and [mathjaxinline]g'(x)[/mathjaxinline]. </p>
<span>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_applications2-tab7-problem2_2_1" class="capa_inputtype">
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<input type="text" name="input_applications2-tab7-problem2_2_1" id="input_applications2-tab7-problem2_2_1" aria-describedby="answer_applications2-tab7-problem2_2_1" value="" style="display:none;"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_applications2-tab7-problem2_2_1" class="answer"/>
</div>
</div></div>
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</span>
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<span id="solution_applications2-tab7-problem2_solution_1"/>
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<h2 class="hd hd-2 unit-title">8. Weighted average</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab8-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab8-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="e43c4aa9fe2511ee9b6e16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
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<h3 class="hd hd-2">Weighted average</h3>
<div
id="video_applications2-tab8-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab8-video1/handler/publish_completion", "streams": "1.00:p4Vwlr8edPg", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab8-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab8-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications2-tab8-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
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<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_applications2-tab8-video1">Downloads and transcripts</h3>
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<h4 class="hd hd-5">Transcripts</h4>
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<p><b class="bfseries">Weighted average</b></p><p>
The <span style="color:#27408C"><b class="bf">weighted average</b></span> value of a function [mathjaxinline]\ f(x)[/mathjaxinline], weighted by a weight function [mathjaxinline]\, w(x)[/mathjaxinline] on an interval [mathjaxinline]a \leq x \leq b[/mathjaxinline] is given by </p><table id="a0000001466" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{\int _ a^ b f(x)w(x)\, dx}{\int _ a^ b w(x)\, dx}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Compute the weighted average of the function [mathjaxinline]\ f(x) = \sin ^2 x[/mathjaxinline] weighted by [mathjaxinline]w(x) = \cos (x)[/mathjaxinline] over the interval [mathjaxinline]0 \leq x \leq \pi /4[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">9. Center of mass</h2>
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<p><h3>1-dimensional center of mass</h3></p><p>
For a 1-dimensional rod of length [mathjaxinline]L[/mathjaxinline] with linear density function [mathjaxinline]\lambda (x)[/mathjaxinline] (measured in mass per unit length), the center of mass is the weighted average: </p><table id="a0000001470" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \bar{x} = \frac{\int _0^ L x \lambda (x)\, dx}{\int _0^ L \lambda (x)\, dx}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Compute the center of mass of a thin rod of length 1 meter with linear density given by [mathjaxinline]\lambda (x) = 10x^2[/mathjaxinline], where [mathjaxinline]\lambda (x)[/mathjaxinline] has units of grams per meter, [mathjaxinline]x[/mathjaxinline] has units of meters, and the constant [mathjaxinline]10[/mathjaxinline] has units of grams per meter cubed. </p>
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<h3 class="hd hd-2">Center of mass example</h3>
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<p><h3>2-dimensional center of mass</h3></p><p>
For a 2-dimensional region of uniform density (mass is proportional to area), the center of mass, or centroid [mathjaxinline](\bar{x}, \bar{y})[/mathjaxinline], is the weighted average of the position with respect to the area. The [mathjaxinline]x[/mathjaxinline]- and [mathjaxinline]y[/mathjaxinline]- coordinates of the centroid are computed by the following integrals: </p><table id="a0000001474" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001475"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \bar{x}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle = \frac{\int _ a^ b x \, dA}{\int _ a^ b dA}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \qquad dA = y\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.15)</td></tr><tr id="a0000001476"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \bar{y}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle = \frac{\int _ a^ b y\, dA}{\int _ a^ b dA}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \qquad dA = x\, dy[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.16)</td></tr></table><p>
We must define the limits of integration so that we integrate over the entire region with respect to [mathjaxinline]x[/mathjaxinline] or [mathjaxinline]y[/mathjaxinline] respectively, and [mathjaxinline]dA[/mathjaxinline] is interpreted as the area of the infinitesimal vertical or horizontal rectangle respectively. </p>
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Center of mass 2
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<p>
Compute the center of mass [mathjaxinline](\bar{x}, \bar{y})[/mathjaxinline] of the unit semi circle centered at the origin, [mathjaxinline]x^2+y^2 \leq 1[/mathjaxinline] and [mathjaxinline]y\geq 0[/mathjaxinline], with uniform density. </p>
<p>
<p style="display:inline">[mathjaxinline]\bar{x}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">11. Worked example</h2>
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<p>
In the cauldron example, heat energy is transferred to the water in the cauldron. <span style="color:#99182C"><b class="bf">Heat</b></span> is defined as the energy required to raise the temperature of an object. So we want to measure the heat required to change the temperature of the water. How do we do this? </p><table id="a0000001485" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \underbrace{\text {Heat energy}}_{\text {cal}} = \underbrace{\text {(Volume)}}_{\text {mL}} \cdot \underbrace{\text {(Temperature)}}_{{}^{\circ {}} \text {Celsius}}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Some units needed in this problem: </p><ul class="itemize"><li><p>
A <span style="color:#99182C"><b class="bf">calorie</b></span> is the amount of energy required to heat one gram of water 1 degree Celsius. </p></li><li><p>
A <span style="color:#99182C"><b class="bf">kilocalorie</b></span> is the amount of energy required to heat one kilogram of water 1 degree Celsius. </p></li><li><p>
One <span style="color:#99182C"><b class="bf">milliliter</b></span> of water weighs one gram. </p></li><li><p>
One <span style="color:#99182C"><b class="bf">cubic centimeter</b></span> is one <span style="color:#99182C"><b class="bf">milliliter</b></span>. </p></li></ul><p><h3>What is Halloween?</h3></p><p>
Halloween is a US holiday on October 31 where kids dress up in costumes and go door to door asking for candy via the phrase “trick or treat" implying they will either take a treat or trick the person. </p>
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<h3 class="hd hd-2">Cauldron example explanation</h3>
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Compute the total heat energy
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In the video, we set up the integral for computing the total heat energy, which is given by </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _0^1 T\cdot \pi x^2 \, dy = \int _0^1 \left(100-30y\right)\cdot \pi y\, dy.[/mathjax]</td>
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Compute this integral. </p>
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Compute the average temperature in the cauldron whose height is 1 meter, whose final temperature is given by [mathjaxinline]T=100-30y[/mathjaxinline] degrees Celsius. (Same set up as above.) </p>
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<span class="trailing_text" id="trailing_text_applications2-tab11-problem2_2_1">degrees Celsius</span>
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<h2 class="hd hd-2 unit-title">12. Work, heat, and Energy</h2>
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<p><span style="color:#27408C"><b class="bf">Work</b></span> is done on an object when <span style="color:#27408C"><b class="bf">energy</b></span> is transferred to that object. </p><p>
In the cauldron example, heat energy was transferred to the water in the cauldron. We were able to measure the work done, or the energy transferred, by measuring the total change in temperature. </p><p><span style="color:#27408C"><b class="bf">Work</b></span>, <span style="color:#27408C"><b class="bf">heat</b></span>, and <span style="color:#27408C"><b class="bf">energy</b></span> have the same units, but we typically measure them differently. </p><ul class="itemize"><li><p>
We measure <span style="color:#27408C"><b class="bf">work</b></span> in Newton-meters (N[mathjaxinline]\cdot[/mathjaxinline]m). </p></li><li><p>
We measure <span style="color:#27408C"><b class="bf">heat</b></span> in calories (cal). </p></li><li><p>
And we measure <span style="color:#27408C"><b class="bf">energy</b></span> in Joules (J). </p></li></ul><p>
James Joule recognized and measured the relationship between heat energy and mechanical work. </p><table id="a0000001491" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle 1 \text { cal} =4.186 \text { J} = 4.186 \text { N}\cdot \text {m}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
The approach to solving problems involving work, heat, or energy is the same, and involves computing an integral. </p><p>
A common example of mechanical work is when a force is applied over a distance. This force applied to the object transfers energy in the form of kinetic or potential energy. In this case, the formula for work is given by </p><table id="a0000001492" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle W = \int F \, dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In the next problems, you will be asked to use integral computations to compute work by thinking of the integral as a cumulative sum. Hint: think about units! </p><p>
Rather than use the Newton-meter, we will be using the English units of foot-pounds (ft[mathjaxinline]\cdot[/mathjaxinline]lbf) for students who have not seen the Newton before. Here, [mathjaxinline]\, 1 \text {lbf}[/mathjaxinline] is a unit of force and equals the weight of a 1 pound mass: [mathjaxinline]\, 1 \, \text {lbf}\, = 1\text {lb}\cdot 9.8 \text {m}/\text {s}^2.\, \,[/mathjaxinline] Below is the conversion between the two systems of units: </p><table id="a0000001493" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle 1 \text {ft}\cdot \text {lbf} \, = \, 1 \text {ft}\cdot \text {lb}\cdot 9.8 \frac{\text {m}}{\text {s}^2} \cdot \frac{1\text { kg}}{2.20462\text { lb}} \cdot \frac{1\text { m}}{3.28084\text { ft}} \, =\, 1.35 \frac{\text {kg}\cdot \text {m}^2}{\text {s}^2} \, =\, 1.35 \text { N}\cdot \text {m}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Consider a spring. According to Hooke's law, the force [mathjaxinline]F[/mathjaxinline] exerted by the spring is given by [mathjaxinline]F=-kx[/mathjaxinline], where [mathjaxinline]k&gt;0[/mathjaxinline] is a positive constant, known as the spring constant, and [mathjaxinline]x[/mathjaxinline] is the displacement from the position at rest ([mathjaxinline]x=0[/mathjaxinline]). </p>
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A force of [mathjaxinline]2,000[/mathjaxinline] lbf moves an extremely stiff spring [mathjaxinline]1/2[/mathjaxinline] inch (in) away from the resting position. How many foot-pounds (ft[mathjaxinline]\cdot[/mathjaxinline]lbf) of work [mathjaxinline]W[/mathjaxinline] would be done in stretching it [mathjaxinline]6[/mathjaxinline] inches from rest? (Note there are 12 inches in 1 foot.) </p>
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<p style="display:inline">[mathjaxinline]W=[/mathjaxinline]</p>
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Work practice 2 (*)
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A heavy-duty rubber firehose hanging over the side of a building is [mathjaxinline]50[/mathjaxinline] feet long and weighs [mathjaxinline]2[/mathjaxinline] lbf/ft. A motorized cylinder (called a winch) winds the hose up until it is on the top of the building. How much work [mathjaxinline]W[/mathjaxinline] (in ft[mathjaxinline]\cdot[/mathjaxinline]lbf) is done by the winch to the hose? </p>
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Ignore the winch, worry only about the vertical force needed to lift the hose. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle W=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">13. Summary</h2>
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<p><b class="bfseries">Average value of f(x)</b></p><p>
The average value of a function [mathjaxinline]\ f(x)[/mathjaxinline] on an interval [mathjaxinline]a \leq x \leq b[/mathjaxinline] is given by </p><table id="a0000001497" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{1}{b-a}\int _ a^ b f(x)\, dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Average value and average rate of change</b></p><p>
Fundamental Theorem of calculus tells us that the average rate of change of [mathjaxinline]y[/mathjaxinline] over [mathjaxinline]a \leq t \leq b[/mathjaxinline] is the <b class="bf">same</b> as the average value of [mathjaxinline]y'[/mathjaxinline] over [mathjaxinline]a \leq t \leq b[/mathjaxinline]. </p><table id="a0000001498" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \text {ave}(y') = \frac{1}{b-a}\int _ a^ b y'(t)\, dt = \frac{y(b)-y(a)}{b-a} = \text {average rate of change of} y(t).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Weighted average</b></p><p>
The <span style="color:#27408C"><b class="bf">weighted average</b></span> value of a function [mathjaxinline]\ f(x)[/mathjaxinline], weighted by a weight function [mathjaxinline]\, w(x)[/mathjaxinline] on an interval [mathjaxinline]a \leq x \leq b[/mathjaxinline] is given by </p><table id="a0000001499" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{\int _ a^ b f(x)w(x)\, dx}{\int _ a^ b w(x)\, dx}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Center of mass</b></p><p><h3>[mathjaxinline]1-[/mathjaxinline]dimensional</h3></p><p>
For a 1-dimensional rod of length [mathjaxinline]L[/mathjaxinline] with linear density function [mathjaxinline]\lambda (x)[/mathjaxinline] (measured in mass per unit length), the center of mass is the weighted average: </p><table id="a0000001500" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle x_ c = \frac{\int _0^ L x \lambda (x)\, dx}{\int _0^ L \lambda (x)\, dx}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><h3>[mathjaxinline]2-[/mathjaxinline]dimensional</h3></p><p>
For a 2-dimensional region whose mass is proportional to area, the center of mass, or centroid [mathjaxinline](\bar{x}, \bar{y})[/mathjaxinline], is the weighted average of the position with respect to the area. The [mathjaxinline]x[/mathjaxinline]- and [mathjaxinline]y[/mathjaxinline]- coordinates of the centroid are computed by the following integrals: </p><table id="a0000001501" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001502"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \bar{x}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{\int _ a^ b x \, dA}{\int _ a^ b dA}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.22)</td></tr><tr id="a0000001503"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \bar{y}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{\int _ a^ b y\, dA}{\int _ a^ b dA},[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.23)</td></tr></table><p>
where [mathjaxinline]dA[/mathjaxinline] is the infinitesimal area, and we must define the limits of integration so that we integrate over the entire region with respect to [mathjaxinline]x[/mathjaxinline] or [mathjaxinline]y[/mathjaxinline] respectively. </p><p><b class="bfseries">Work, heat, and energy</b></p><p><span style="color:#27408C"><b class="bf">Work</b></span> is done on an object when <span style="color:#27408C"><b class="bf">energy</b></span> is transferred to that object. </p><p><span style="color:#27408C"><b class="bf">Work</b></span>, <span style="color:#27408C"><b class="bf">heat</b></span>, and <span style="color:#27408C"><b class="bf">energy</b></span> have the same units, but we typically measure them differently. </p><ul class="itemize"><li><p>
We measure <span style="color:#27408C"><b class="bf">work</b></span> in Newton-meters (N[mathjaxinline]\cdot[/mathjaxinline]m). </p></li><li><p>
We measure <span style="color:#27408C"><b class="bf">heat</b></span> in calories (cal). </p></li><li><p>
And we measure <span style="color:#27408C"><b class="bf">energy</b></span> in Joules (J). </p></li></ul><p>
James Joule recognized and measured the relationship between heat energy and mechanical work. </p><table id="a0000001504" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle 1 \text {cal} =4.186 \text { J} = 4.186 \text { N}\cdot \text {m}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
The approach to solving problems involving work, heat, or energy is the same, and involves computing an integral. </p><p>
A common example of mechanical work is when a force is applied over a distance. This force applied to the object transfers energy in the form of kinetic or potential energy. In this case, the formula for work is given by </p><table id="a0000001505" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle W = \int F \, dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In the cauldron example, heat energy was transferred to the water in the cauldron. We measured the work done, or the energy transferred, by measuring the total change in temperature [mathjaxinline]T[/mathjaxinline]: </p><table id="a0000001506" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int T dV.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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