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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Areas and volumes</h3>
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<h2 class="hd hd-2 unit-title">2. Areas and volumes</h2>
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<p><h3>Objectives</h3></p><ul class="itemize"><li><p>
Apply Riemann integration to find <span style="color:#99182C"><b class="bf">areas bounded between curves</b></span>. </p></li><li><p>
Shift perspective between integrating with respect to [mathjaxinline]x[/mathjaxinline] and integrating with respect to [mathjaxinline]y[/mathjaxinline] depending on problem geometry. </p></li><li><p>
Apply Riemann integration to find <span style="color:#99182C"><b class="bf">volumes</b></span> of shapes whose cross-sectional area is known. </p></li><li><p>
Determine <span style="color:#99182C"><b class="bf">limits of integration</b></span>, and use the <span style="color:#99182C"><b class="bf">method of disks</b></span>, <span style="color:#99182C"><b class="bf">washers</b></span>, or <span style="color:#99182C"><b class="bf">shells</b></span> to determine <span style="color:#99182C"><b class="bf">volumes of revolution</b></span>. </p></li></ul><p><h3>Contents: 17 pages</h3></p><p>
13 videos (82 minutes at 1x speed) 24 problems </p>
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<h2 class="hd hd-2 unit-title">3. Preparation</h2>
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Preparation
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<p>
We can approximate the area between curves using Riemann sums of rectangles. </p>
<center><img alt="Rectangles approximating the area between a region bounded by top curve f of x and bottom curve g of x" src="/assets/courseware/v1/98926bd0853b04520b92336a2bd7f797/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u7s1_areasbetweencurves_2.svg" style="margin: 10px 25px 25px 25px" width="300px"/>Two functions f of x and g of x are plotted in the first quadrant of the x y plane. The function f of x is greater than g of x for every x. The area between the two curves is approximated by five rectangles side-by-side. </center>
<p>
Which differential(s) describes the area of the infinitesimal rectangle of width [mathjaxinline]dx[/mathjaxinline] in the image below? </p>
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<text>[mathjaxinline]\left(f(x)+g(x)\right)dx[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">4. Areas between curves</h2>
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Drawing the region
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Draw the region whose area is bounded between the curves [mathjaxinline]y=2x^2[/mathjaxinline] and [mathjaxinline]y=x^2[/mathjaxinline] over the interval [mathjaxinline]0\leq x \leq 1[/mathjaxinline]. </p>
<p>
(Use the Top Function tool to draw the curve that bounds the area from above on the region, and use the Bottom Function tool to draw the curve that bounds the region from below.) </p>
<font size="2">Note that when drawing the function, you may choose between using the freeform or spline drawing tool using the dropdown menu in the menu of the sketch response tool. The freeform tool draws like a regular pencil using your mouse. The spline tool allows you to pick a series of discrete points, and it then connects them (once you have two or more points) with a curve. </font>
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Limits of integration
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Express the area of the region (as in the problem above) bounded by the curves [mathjaxinline]y=2x^2[/mathjaxinline] and [mathjaxinline]y=x^2[/mathjaxinline] over the interval [mathjaxinline]0\leq x \leq 1[/mathjaxinline] as an integral. Then compute the area. </p>
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<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
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<span class="trailing_text" id="trailing_text_applications1-tab4-problem2_3_1"> [mathjaxinline] dx[/mathjaxinline]</span>
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\(\)
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
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<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<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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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Drawing the region
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<p>
Draw the region whose area is bounded above by the line [mathjaxinline]y=1,\,[/mathjaxinline] and bounded below by the curves [mathjaxinline]y=\sqrt {x}[/mathjaxinline] for [mathjaxinline]x\geq 0[/mathjaxinline], and [mathjaxinline]y=x^2[/mathjaxinline] for [mathjaxinline]x\leq 0[/mathjaxinline]. </p>
<p>
(Use the <b class="bfseries"><span style="color:#0000FF">Top function</span></b> tool to draw the curve that bounds the area from above on the region, and use the Bottom function tool to draw the curve that bounds the region from below. Use the <b class="bf">Intersection points</b> tool to mark intersection points between any of the 3 curves. Use the <b class="bf">Vertical line</b> tool to drop a line indicating where to split the region into two integrals. ) </p>
<font size="2">Note that when drawing the function, you may choose between using the freeform or spline drawing tool using the dropdown menu in the menu of the sketch response tool. The freeform tool draws like a regular pencil using your mouse. The spline tool allows you to pick a series of discrete points, and it then connects them (once you have two or more points) with a curve. </font>
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Find the area
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Find the area of the region bounded above by the line [mathjaxinline]y=1,\,[/mathjaxinline] and bounded below by the curves [mathjaxinline]y=\sqrt {x}[/mathjaxinline] for [mathjaxinline]x\geq 0[/mathjaxinline], and [mathjaxinline]y=x^2[/mathjaxinline] for [mathjaxinline]x\leq 0[/mathjaxinline]. (Same region as you drew above.) </p>
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<h2 class="hd hd-2 unit-title">6. More areas between curves</h2>
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<h3 class="hd hd-2">Example video: area of between graphs of sine and cosine</h3>
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<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab6-video2/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
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<h2 class="hd hd-2 unit-title">7. Changing direction</h2>
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Preparation
</h3>
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<div class="problem">
<div>
<center>
<img alt="Two functions f of x and g of x are plotted in the first quadrant of the x y plane. The functions intersect at two points. Between the points of intersection, the function f of x is less than g of x. A thin rectangle is shown with height dy, left end touching f of x, and right end touching g of x." src="/assets/courseware/v1/c20c64c27ffe360acfd40a0de9308610/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u7s1_areasbetweencurves_1_v3.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
Which differential describes the area of the infinitesimal rectangle of height [mathjaxinline]dy[/mathjaxinline] in the image above? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_applications1-tab7-problem1_2_1">
<fieldset aria-describedby="status_applications1-tab7-problem1_2_1">
<div class="field">
<input type="radio" name="input_applications1-tab7-problem1_2_1" id="input_applications1-tab7-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="applications1-tab7-problem1_2_1-choice_1-label" for="input_applications1-tab7-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_applications1-tab7-problem1_2_1">
<text> [mathjaxinline]\left(f(x)+g(x)\right)dx[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_applications1-tab7-problem1_2_1" id="input_applications1-tab7-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="applications1-tab7-problem1_2_1-choice_2-label" for="input_applications1-tab7-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_applications1-tab7-problem1_2_1">
<text> [mathjaxinline]\left(f(x)-g(x)\right)dx[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_applications1-tab7-problem1_2_1" id="input_applications1-tab7-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="applications1-tab7-problem1_2_1-choice_3-label" for="input_applications1-tab7-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_applications1-tab7-problem1_2_1">
<text> [mathjaxinline]\left(g(x)-f(x)\right)dx[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_applications1-tab7-problem1_2_1" id="input_applications1-tab7-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="applications1-tab7-problem1_2_1-choice_4-label" for="input_applications1-tab7-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_applications1-tab7-problem1_2_1">
<text> [mathjaxinline]\left(f^{-1}(y)-g^{-1}(y)\right)dy[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_applications1-tab7-problem1_2_1" id="input_applications1-tab7-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="applications1-tab7-problem1_2_1-choice_5-label" for="input_applications1-tab7-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_applications1-tab7-problem1_2_1">
<text> [mathjaxinline]\left(g^{-1}(y)-f^{-1}(y)\right)dy[/mathjaxinline]</text>
</label>
</div>
<span id="answer_applications1-tab7-problem1_2_1"/>
</fieldset>
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<span class="status unanswered" id="status_applications1-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>
</p>
<p>
<div class="solution-span">
<span id="solution_applications1-tab7-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
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"
data-graded="True">
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<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
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<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab7-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-request-token="38d07264ed8d11eeb91e16ffd59041e1" data-graded="True" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab7-video1" data-block-type="video" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Shifting perspective</h3>
<div
id="video_applications1-tab7-video1"
class="video closed"
data-metadata='{"completionPercentage": 0.95, "ytMetadataEndpoint": "", "prioritizeHls": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab7-video1/handler/transcript/available_translations", "generalSpeed": 1.0, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab7-video1/handler/xmodule_handler/save_user_state", "savedVideoPosition": 0.0, "ytApiUrl": "https://www.youtube.com/iframe_api", "end": 0.0, "poster": null, "ytTestTimeout": 1500, "completionEnabled": false, "streams": "1.00:uQScaJhzvYc", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab7-video1/handler/publish_completion", "showCaptions": "true", "start": 0.0, "sources": [], "autoplay": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab7-video1/handler/transcript/translation/__lang__", "autoAdvance": false, "autohideHtml5": false, "lmsRootURL": "https://openlearninglibrary.mit.edu", "speed": null, "transcriptLanguage": "en", "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
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<div id="applications1-tab7-video1"></div>
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Your browser does not support this video format. Try using a different browser.
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<div class="video-player-post"></div>
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<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_applications1-tab7-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_applications1-tab7-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab7-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@applications1-tab7-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
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<h2 class="hd hd-2 unit-title">8. Shifting perspectives practice</h2>
<div class="vert-mod">
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aria-labelledby="applications1-tab8-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications1-tab8-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications1-tab8-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="8"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="applications1-tab8-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications1-tab8-problem1-problem-progress" tabindex="-1">
Shifting perspectives
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications1-tab8-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Find the area bounded by the curves [mathjaxinline]y = 1[/mathjaxinline] and [mathjaxinline]y=x^2[/mathjaxinline] in two ways. (Do step one of drawing the region on your own.) </p>
<p>
First, find the area [mathjaxinline]A[/mathjaxinline] by integrating over [mathjaxinline]x[/mathjaxinline]: </p>
<span>
<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style>
<table>
<tbody>
<tr>
<td colspan="3">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_applications1-tab8-problem1_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_applications1-tab8-problem1_2_1" id="input_applications1-tab8-problem1_2_1" aria-describedby="status_applications1-tab8-problem1_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_applications1-tab8-problem1_2_1"/>
<span class="status unanswered" id="status_applications1-tab8-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab8-problem1_2_1" class="answer"/>
</div>
</div></div>
</td>
</tr>
<tr>
<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
</td>
<td>
<br/>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_applications1-tab8-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_applications1-tab8-problem1_3_1" id="input_applications1-tab8-problem1_3_1" data-input-id="applications1-tab8-problem1_3_1" value="" aria-describedby="trailing_text_applications1-tab8-problem1_3_1 status_applications1-tab8-problem1_3_1" size="30"/>
<span class="trailing_text" id="trailing_text_applications1-tab8-problem1_3_1"> [mathjaxinline] dx[/mathjaxinline]</span>
<span class="status unanswered" id="status_applications1-tab8-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab8-problem1_3_1" class="answer"/>
<div id="input_applications1-tab8-problem1_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</td>
<td>
<p style="display:inline"> \( = \)</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_applications1-tab8-problem1_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_applications1-tab8-problem1_4_1" id="input_applications1-tab8-problem1_4_1" aria-describedby="status_applications1-tab8-problem1_4_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_applications1-tab8-problem1_4_1"/>
<span class="status unanswered" id="status_applications1-tab8-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab8-problem1_4_1" class="answer"/>
</div>
</div></div>
</td>
</tr>
<tr>
<td colspan="3">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_applications1-tab8-problem1_5_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_applications1-tab8-problem1_5_1" id="input_applications1-tab8-problem1_5_1" aria-describedby="status_applications1-tab8-problem1_5_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_applications1-tab8-problem1_5_1"/>
<span class="status unanswered" id="status_applications1-tab8-problem1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab8-problem1_5_1" class="answer"/>
</div>
</div></div>
</td>
</tr>
</tbody>
</table>
</span>
<p>
Now find the area [mathjaxinline]A[/mathjaxinline] by integrating over [mathjaxinline]y[/mathjaxinline]: </p>
<span>
<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
</style>
<table>
<tbody>
<tr>
<td colspan="3">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_applications1-tab8-problem1_6_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_applications1-tab8-problem1_6_1" id="input_applications1-tab8-problem1_6_1" aria-describedby="status_applications1-tab8-problem1_6_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_applications1-tab8-problem1_6_1"/>
<span class="status unanswered" id="status_applications1-tab8-problem1_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
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<br/>
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<span class="trailing_text" id="trailing_text_applications1-tab8-problem1_7_1"> [mathjaxinline] dy[/mathjaxinline]</span>
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\(\)
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<p style="display:inline"> \( = \)</p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<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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Find the area between the curves [mathjaxinline]y = 4-x^2[/mathjaxinline] and [mathjaxinline]y=3x[/mathjaxinline]. </p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<p><b class="bfseries">Volumes by slicing</b></p><p>
The volume of a slice of bread is </p><p>
[mathjaxinline]dV = A(x)\, dx,[/mathjaxinline] </p><p>
where [mathjaxinline]dx[/mathjaxinline] is the width of the slice, and [mathjaxinline]A(x)[/mathjaxinline] is the cross-sectional area. The volume of the entire loaf is </p><p>
[mathjaxinline]V = \int A(x) \, dx[/mathjaxinline]. </p>
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Pyramid
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<p>
Find the volume of the square bottom pyramid. The height and base width are both [mathjaxinline]a[/mathjaxinline]. The [mathjaxinline]x[/mathjaxinline]-axis passes through the center of the pyramid. </p>
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<img alt="Horizontal pyramid with vertex at origin, square cross section of width a and x equals a" src="/assets/courseware/v1/9cee1862d5a3b7b2524d26385ecdf31d/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_pyramid.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
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<p>
First step, what is the volume of the infinitesimal slice depicted? </p>
<p>
(Enter as a function of [mathjaxinline]x[/mathjaxinline].) </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_applications1-tab9-problem1_2_1" class="inputtype formulaequationinput">
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<span class="trailing_text" id="trailing_text_applications1-tab9-problem1_2_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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\(\)
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<p>
What is the integral expression that computes the volume? Use that integral to compute the volume. </p>
<p>
(Enter limits of integration and integrand including. Answers may involve the parameter [mathjaxinline]a[/mathjaxinline]. All boxes are graded together.) </p>
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<div id="display_applications1-tab9-problem1_3_1" class="equation">`{::}`</div>
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<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
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<br/>
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<span class="trailing_text" id="trailing_text_applications1-tab9-problem1_4_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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\(\)
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<p style="display:inline"> \( = \)</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"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
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<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Use integration to find the volume of the cone whose base has radius 1 and whose height is 2. </p>
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<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput"> 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">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<h2 class="hd hd-2 unit-title">10. Method of disks</h2>
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<p><b class="bfseries">Method of disks</b></p><center><img src="/assets/courseware/v1/532d310da48e95b6b2fb34ca4c02de06/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_methoddisks2d.svg" width="250px" alt="2d rectangle" style="margin: 10px 25px 25px 25px"/> <img src="/assets/courseware/v1/52ae9af21377c80bc262fe9323fc6768/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_methoddisks3d.svg" width="250px" alt="3d rectangle rotated to become a disk" style="margin: 10px 25px 25px 25px"/></center><p>
The method of disks gives the differential volume element for a curve rotated about the [mathjaxinline]x-[/mathjaxinline] axis </p><table id="a0000001424" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]dV = \pi y^2 dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Find the volume of the disk
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Consider the region created by rotating [mathjaxinline]\sin (x)[/mathjaxinline] about the [mathjaxinline]x-[/mathjaxinline]axis for [mathjaxinline]0 \leq x \leq \pi[/mathjaxinline]. </p>
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Find the expression for the volume of an infinitesimal disk of this solid of revolution. </p>
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(Enter expression in terms of [mathjaxinline]x[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]dV=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Example Entries</th>
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<td class="formulainput">Integers</td>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</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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Set up the integral
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Consider the region in the first quadrant bounded by [mathjaxinline]y=x^2[/mathjaxinline] and the line [mathjaxinline]x=1[/mathjaxinline], the [mathjaxinline]x-[/mathjaxinline]axis and the [mathjaxinline]y[/mathjaxinline]-axis. </p>
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<img alt="" src="/assets/courseware/v1/3fba93e57cc6337746319af1fc2f4a7e/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_app1-vol1.svg" style="margin: 10px 25px 25px 25px" width="220px"/>
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Revolve the region about the [mathjaxinline]x[/mathjaxinline]-axis. </p>
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Find the expression for the volume of an infinitesimal disk. (Enter differential expression.) </p>
<p>
<p style="display:inline">[mathjaxinline]dV=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_applications1-tab10-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<span class="trailing_text" id="trailing_text_applications1-tab10-problem2_2_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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\(\)
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Set up the limits of integration, the integrand, and solve for the volume. </p>
<p>
(Enter the smaller integration limit as the lower limit and the larger integration limit as the upper limit. Note that you do not need to enter [mathjaxinline]dx[/mathjaxinline] for the integrand.) <br/></p>
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<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
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<br/>
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<input type="text" name="input_applications1-tab10-problem2_4_1" id="input_applications1-tab10-problem2_4_1" data-input-id="applications1-tab10-problem2_4_1" value="" aria-describedby="trailing_text_applications1-tab10-problem2_4_1 status_applications1-tab10-problem2_4_1" size="30"/>
<span class="trailing_text" id="trailing_text_applications1-tab10-problem2_4_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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\(\)
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<p style="display:inline"> \( = \)</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</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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<h2 class="hd hd-2 unit-title">11. Example: volume of a sphere</h2>
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<h2 class="hd hd-2 unit-title">12. Method of shells</h2>
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Units
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We found by the method of shells that the volume of one shell is given by the formula </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle dV = 2\pi x(a-y)dx.[/mathjax]</td>
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If [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline] have units of meters, what are the units of this differential volume element? </p>
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(Enter m for meters.) </p>
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<p><b class="bfseries">Food for thought</b></p><p>
In the equation [mathjaxinline]y=x^2[/mathjaxinline], if [mathjaxinline]x[/mathjaxinline] has units of meters, how does [mathjaxinline]y[/mathjaxinline] also have units of meters? </p>
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<h3 class="hd hd-2">A paradox of units</h3>
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<h2 class="hd hd-2 unit-title">13. Method of shells practice</h2>
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The method of shells for a curve [mathjaxinline]y=f(x)[/mathjaxinline] rotated about the [mathjaxinline]y[/mathjaxinline]-axis: </p><table id="a0000001428" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]dV = (2\pi x) y \, dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Method of shells practice 1
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In this problem you set up the method of shells by using a horizontal slice revolved about the [mathjaxinline]x[/mathjaxinline]-axis. </p>
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Set up the integrals for the volume of the solid generated by rotating the region bounded by [mathjaxinline]y=\sqrt {a^2-x^2}\, \, (a&gt;0)[/mathjaxinline] and [mathjaxinline]y=0[/mathjaxinline] around the [mathjaxinline]x[/mathjaxinline]-axis by the method of shells. The [mathjaxinline]a[/mathjaxinline] is constant. (Do not evaluate.)<br/></p>
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(Enter the smaller integration limit as the lower limit and the larger integration limit as the upper limit. Note that you do not need to enter the differential [mathjaxinline]dy[/mathjaxinline] for the integrand.) <br/></p>
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<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
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<span class="trailing_text" id="trailing_text_applications1-tab13-problem1_3_1"> [mathjaxinline] dy[/mathjaxinline]</span>
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Method of shells practice 2
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Set up the integrals for the volume of the solid generated by rotating the region bounded by [mathjaxinline]y=x[/mathjaxinline], [mathjaxinline]y=0[/mathjaxinline], and [mathjaxinline]x=1[/mathjaxinline] around the [mathjaxinline]x[/mathjaxinline]-axis by the method of shells. (Do not evaluate.)<br/></p>
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(Enter the smaller integration limit as the lower limit and the larger integration limit as the upper limit. Note that you do not need to enter the differential [mathjaxinline]dy[/mathjaxinline] for the integrand.) <br/></p>
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<span class="trailing_text" id="trailing_text_applications1-tab13-problem2_3_1"> [mathjaxinline] dy[/mathjaxinline]</span>
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Method of shells practice 3
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Set up the integrals for the volume of the solid generated by rotating the region bounded by [mathjaxinline]y=2x-x^2[/mathjaxinline] and [mathjaxinline]y=0[/mathjaxinline] around the [mathjaxinline]y[/mathjaxinline]-axis by the method of shells. (Do not evaluate.)<br/></p>
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(Enter the smaller integration limit as the lower limit and the larger integration limit as the upper limit. Note that you do not need to enter the differential [mathjaxinline]dx[/mathjaxinline] for the integrand.) <br/></p>
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<br/>
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<span class="trailing_text" id="trailing_text_applications1-tab13-problem3_3_1"> [mathjaxinline] dx[/mathjaxinline]</span>
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<h2 class="hd hd-2 unit-title">14. Method of washers</h2>
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Method of washers
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<p>
Consider the region bounded by [mathjaxinline]y=x^2[/mathjaxinline] and the line [mathjaxinline]y=x[/mathjaxinline] rotated about the [mathjaxinline]x[/mathjaxinline]-axis. </p>
<center>
<img alt="" src="/assets/courseware/v1/958163f00909d98ef1e0154b84726212/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_app1_washer3.svg" style="margin: 10px 25px 25px 25px" width="225px"/>
</center>
<p>
The method we introduce in this problem is similar to the method of disks. Instead of a disk shape, we get a disk with another disk cut out of it, sometimes called a washer. </p>
<center>
<img alt="" src="/assets/courseware/v1/7b153d90fc71249301ac3b40406989f3/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_app1_washer2.svg" style="margin: 10px 25px 25px 25px" width="225px"/>
</center>
<p>
We call this the method of washers. </p>
<p>
What is the volume of an infinitesimal disk obtained by rotating [mathjaxinline]y=x^2[/mathjaxinline] about the [mathjaxinline]x-[/mathjaxinline]axis? </p>
<p>
<p style="display:inline">[mathjaxinline]dV_{\text {small disk}}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_applications1-tab14-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_applications1-tab14-problem1_2_1" id="input_applications1-tab14-problem1_2_1" data-input-id="applications1-tab14-problem1_2_1" value="" aria-describedby="trailing_text_applications1-tab14-problem1_2_1 status_applications1-tab14-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_applications1-tab14-problem1_2_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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\(\)
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</p>
<p>
What is the volume of an infinitesimal disk obtained by rotating [mathjaxinline]y=x[/mathjaxinline] about the [mathjaxinline]x-[/mathjaxinline]axis? </p>
<p>
<p style="display:inline">[mathjaxinline]dV_{\text {large disk}}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_applications1-tab14-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
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<span class="trailing_text" id="trailing_text_applications1-tab14-problem1_3_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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</p>
<p>
What is the volume of an infinitesimal washer obtained by rotating the region bounded by [mathjaxinline]y=x^2[/mathjaxinline] and [mathjaxinline]y=x[/mathjaxinline] about the [mathjaxinline]x-[/mathjaxinline]axis? </p>
<p>
<p style="display:inline">[mathjaxinline]dV_{\text {washer}}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_applications1-tab14-problem1_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<span class="trailing_text" id="trailing_text_applications1-tab14-problem1_4_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p>
Compute the volume. </p>
<p>
<p style="display:inline">[mathjaxinline]V=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">15. Method of disks and washers practice</h2>
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Practice 1
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Consider the region bounded by [mathjaxinline]y=x^2[/mathjaxinline] and the line [mathjaxinline]y=x[/mathjaxinline] rotated about the <b class="bf">[mathjaxinline]y[/mathjaxinline]-axis</b>. Find the volume using the method of washers. </p>
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<img alt="" src="/assets/courseware/v1/958163f00909d98ef1e0154b84726212/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_app1_washer3.svg" style="margin: 10px 25px 25px 25px" width="225px"/>
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<p style="display:inline">[mathjaxinline]dV=[/mathjaxinline]</p>
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Compute the volume. </p>
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<p style="display:inline">[mathjaxinline]V=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
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<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<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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<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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<h3 class="hd hd-3 problem-header" id="applications1-tab15-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@applications1-tab15-problem2-problem-progress" tabindex="-1">
Practice 2
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The ball of radius 1 is obtained by rotating the curve [mathjaxinline]y=\sqrt {1-x^2}[/mathjaxinline] about the [mathjaxinline]x-[/mathjaxinline]axis. A cylindrical hole of radius [mathjaxinline]a[/mathjaxinline] is drilled out of the ball (obtained by rotating the horizontal line [mathjaxinline]y=a[/mathjaxinline] about the [mathjaxinline]x-[/mathjaxinline]axis). </p>
<center>
<img alt="" src="/assets/courseware/v1/170c446ffadc0a65a325f6b30d131f04/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_app1_ball_cylinder2.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
</center>
<p>
Use the method of washers to find set up the volume of the ball of radius 1 with a cylinder of radius [mathjaxinline]a&lt;1[/mathjaxinline] removed. </p>
<p>
(Enter integrand but not the infinitesimal [mathjaxinline]dx[/mathjaxinline]. All three answer boxes are graded together.) </p>
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<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
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<td>
<br/>
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<span class="trailing_text" id="trailing_text_applications1-tab15-problem2_2_2"> [mathjaxinline] dx[/mathjaxinline]</span>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Revolve around both axes
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Use the method of disks to find the volume [mathjaxinline]V_ x[/mathjaxinline]of the solid of revolution generated by rotating the regions bounded by the curves [mathjaxinline]\displaystyle y=2x-x^2[/mathjaxinline] and [mathjaxinline]\displaystyle y=0[/mathjaxinline] around the [mathjaxinline]x[/mathjaxinline]-axis. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle V_ x=[/mathjaxinline]</p>
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<p>
Use the method of washers to find the volume [mathjaxinline]V_ y[/mathjaxinline]of the solid of revolution generated by rotating the regions bounded by the curves [mathjaxinline]\displaystyle y=2x-x^2[/mathjaxinline] and [mathjaxinline]\displaystyle y=0[/mathjaxinline] around the [mathjaxinline]y[/mathjaxinline]-axis. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle V_ y=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">16. Rotating about an arbitrary axis</h2>
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Practice 1
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Let [mathjaxinline]0&lt;a&lt;b[/mathjaxinline]. Consider a ball of radius [mathjaxinline]b[/mathjaxinline] and with a cylindrical hole drilled out of it of radius [mathjaxinline]a[/mathjaxinline] whose axis passes through the center of the ball. </p>
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<img alt="" src="/assets/courseware/v1/cb55e2e521ecc1b93d9c29db062a6697/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_app1_ball_cylinder.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
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<p>
Use the method of shells to find the volume [mathjaxinline]V[/mathjaxinline] of the ball (in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]) with the cylinder removed. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle V=[/mathjaxinline]</p>
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<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
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Practice 2
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<p>
Find the volume of the region [mathjaxinline]\sqrt x \le y \le 1[/mathjaxinline], [mathjaxinline]x\ge 0[/mathjaxinline] revolved around the [mathjaxinline]y[/mathjaxinline]-axis by both the method of shells and the method of disks and washers. </p>
<p>
Shells: <span><style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style><table><tbody><tr><td colspan="4"><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_applications1-tab16-problem2_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_applications1-tab16-problem2_2_1" id="input_applications1-tab16-problem2_2_1" aria-describedby="status_applications1-tab16-problem2_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_applications1-tab16-problem2_2_1"/>
<span class="status unanswered" id="status_applications1-tab16-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab16-problem2_2_1" class="answer"/>
</div>
</div></div></td></tr><tr><td><p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p></td><td><br/><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_applications1-tab16-problem2_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_applications1-tab16-problem2_3_1" id="input_applications1-tab16-problem2_3_1" data-input-id="applications1-tab16-problem2_3_1" value="" aria-describedby="trailing_text_applications1-tab16-problem2_3_1 status_applications1-tab16-problem2_3_1" size="30"/>
<span class="trailing_text" id="trailing_text_applications1-tab16-problem2_3_1"> [mathjaxinline] dx[/mathjaxinline]</span>
<span class="status unanswered" id="status_applications1-tab16-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab16-problem2_3_1" class="answer"/>
<div id="input_applications1-tab16-problem2_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div></td><td><p style="display:inline; text-align:right"> \( = \)</p></td><td><div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_applications1-tab16-problem2_4_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_applications1-tab16-problem2_4_1" id="input_applications1-tab16-problem2_4_1" data-input-id="applications1-tab16-problem2_4_1" value="" aria-describedby="status_applications1-tab16-problem2_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_applications1-tab16-problem2_4_1"/>
<span class="status unanswered" id="status_applications1-tab16-problem2_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab16-problem2_4_1" class="answer"/>
<div id="input_applications1-tab16-problem2_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div></td></tr><tr><td colspan="4"><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_applications1-tab16-problem2_5_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_applications1-tab16-problem2_5_1" id="input_applications1-tab16-problem2_5_1" aria-describedby="status_applications1-tab16-problem2_5_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_applications1-tab16-problem2_5_1"/>
<span class="status unanswered" id="status_applications1-tab16-problem2_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab16-problem2_5_1" class="answer"/>
</div>
</div></div></td></tr></tbody></table></span></p>
<p>
Disks: <span><style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style><table><tbody><tr><td colspan="4"><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_applications1-tab16-problem2_6_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_applications1-tab16-problem2_6_1" id="input_applications1-tab16-problem2_6_1" aria-describedby="status_applications1-tab16-problem2_6_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_applications1-tab16-problem2_6_1"/>
<span class="status unanswered" id="status_applications1-tab16-problem2_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab16-problem2_6_1" class="answer"/>
</div>
</div></div></td></tr><tr><td><p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p></td><td><br/><div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="formulaequationinput_applications1-tab16-problem2_7_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_applications1-tab16-problem2_7_1" id="input_applications1-tab16-problem2_7_1" data-input-id="applications1-tab16-problem2_7_1" value="" aria-describedby="trailing_text_applications1-tab16-problem2_7_1 status_applications1-tab16-problem2_7_1" size="30"/>
<span class="trailing_text" id="trailing_text_applications1-tab16-problem2_7_1"> [mathjaxinline] dy[/mathjaxinline]</span>
<span class="status unanswered" id="status_applications1-tab16-problem2_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab16-problem2_7_1" class="answer"/>
<div id="input_applications1-tab16-problem2_7_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div></td><td><p style="display:inline; text-align:right"> \( = \)</p></td><td><div class="inline" tabindex="-1" aria-label="Question 7" role="group"><div id="formulaequationinput_applications1-tab16-problem2_8_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_applications1-tab16-problem2_8_1" id="input_applications1-tab16-problem2_8_1" data-input-id="applications1-tab16-problem2_8_1" value="" aria-describedby="status_applications1-tab16-problem2_8_1" size="20"/>
<span class="trailing_text" id="trailing_text_applications1-tab16-problem2_8_1"/>
<span class="status unanswered" id="status_applications1-tab16-problem2_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab16-problem2_8_1" class="answer"/>
<div id="input_applications1-tab16-problem2_8_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div></td></tr><tr><td colspan="3"><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 8" role="group"><div id="inputtype_applications1-tab16-problem2_9_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_applications1-tab16-problem2_9_1" id="input_applications1-tab16-problem2_9_1" aria-describedby="status_applications1-tab16-problem2_9_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_applications1-tab16-problem2_9_1"/>
<span class="status unanswered" id="status_applications1-tab16-problem2_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_applications1-tab16-problem2_9_1" class="answer"/>
</div>
</div></div></td></tr></tbody></table></span></p>
<span>
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<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Revolve an equilateral triangle
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Find the volume [mathjaxinline]V[/mathjaxinline] of the solid obtained by revolving an equilateral triangle of side-length [mathjaxinline]a[/mathjaxinline] around one of its sides. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle V=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<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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<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<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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Rotation about a different axis
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Consider the region bounded by [mathjaxinline]y=x^2[/mathjaxinline] and [mathjaxinline]y=x[/mathjaxinline] rotated about the line [mathjaxinline]x=2[/mathjaxinline]. </p>
<p>
Find the differential volume by the method of shells. </p>
<p>
(Enter in terms of [mathjaxinline]x[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]dV=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_applications1-tab16-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<span class="trailing_text" id="trailing_text_applications1-tab16-problem4_2_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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Find the differential volume by the method of disks or washers. </p>
<p>
(Enter in terms of [mathjaxinline]y[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]dV=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_applications1-tab16-problem4_3_1"> [mathjaxinline]dy[/mathjaxinline]</span>
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Compute the volume by method of your choice. <p style="display:inline">[mathjaxinline]V=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_applications1-tab16-problem4_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</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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<h2 class="hd hd-2 unit-title">17. Summary</h2>
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<p><b class="bfseries">Process for finding areas and volumes</b></p><ol class="enumerate"><li value="1"><p>
Draw a picture </p></li><li value="2"><p>
Identify limits of integration </p></li><li value="3"><p>
Identify integrand </p></li><li value="4"><p>
Integrate to solve (as needed) </p></li></ol><p><b class="bfseries">Volumes by slicing</b></p><p>
The volume of a slice of bread is </p><p>
[mathjaxinline]dV = A(x)\, dx,[/mathjaxinline] </p><p>
where [mathjaxinline]dx[/mathjaxinline] is the width of the slice, and [mathjaxinline]A(x)[/mathjaxinline] is the cross-sectional area. The volume of the entire loaf is </p><p>
[mathjaxinline]V = \int A(x) \, dx.[/mathjaxinline] </p><p><b class="bfseries">Method of disks</b></p><center><img src="/assets/courseware/v1/532d310da48e95b6b2fb34ca4c02de06/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_methoddisks2d.svg" width="250px" alt="2d rectangle area element" style="margin: 10px 25px 25px 25px"/> <img src="/assets/courseware/v1/52ae9af21377c80bc262fe9323fc6768/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_methoddisks3d.svg" width="250px" alt="2d disk, volume element" style="margin: 10px 25px 25px 25px"/></center><p>
The method of disks gives the differential volume element for a curve rotated about the [mathjaxinline]x-[/mathjaxinline] axis </p><table id="a0000001450" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]dV = \pi y^2\, dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Method of shells</b></p><center><img src="/assets/courseware/v1/4056922b360c2727f5c313404d0f7fc8/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_methodshells.svg" width="300px" alt="" style="margin: 10px 25px 25px 25px"/></center><p>
The method of shells for a curve [mathjaxinline]y=f(x)[/mathjaxinline] rotated about the [mathjaxinline]y[/mathjaxinline]-axis: </p><table id="a0000001451" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]dV = 2\pi x y\, dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Method of washers</b></p><center><img src="/assets/courseware/v1/7b153d90fc71249301ac3b40406989f3/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_app1_washer2.svg" width="225px" alt="" style="margin: 10px 25px 25px 25px"/></center><p>
The volume of an infinitesimal washer is given by </p><table id="a0000001452" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]dV = \pi (y_{\textrm{outer}}^2 - y_{\textrm{inner}}^2)\, dx.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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