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src="../../files/integration/volume/main.png" style="float: left; width: 100px; height: 100px;"> In this page, we will learn about how to find the volume generated by rotating a region around the x axis and the y axis. The formule for these are not difficult to use. The difficulty often comes with applying the integration techniques. It is therefore recommended that you revise these techniques before you go through this topic. In particular, you should be confident with <a href="../634/integration-by-substitution-hl.html" title="Integration by Substitution HL">Integration by Substitution</a> and <a href="../635/integration-by-parts.html" title="Integration by Parts">Integration by Parts</a>.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about</p> <ul> <li>finding the volume generated by rotating a region under a curve about the x axis/y axis</li> <li>finding the volume generated by rotating a region bounded by two graphs about the x axis/y axis</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe frameborder="0" height="480" scrolling="no" src="../../files/integration/volume/revision-notes_volume-of-revolution.pdf" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <hr class="hidden-separator"> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#60a73ede"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="60a73ede"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Volume of Revolution <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-919-644" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve y = f(x)</p><p>The region bounded by the curve and the lines y = 0 and x = a and x = b is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> x axis</strong>.</p><p>Which of the following represents the volume of this solid?</p><p><img alt="" src="../../files/integration/volume/quiz/q1.png" style="width: 450px; height: 334px;"></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large V=\pi\int_{a}^{b}[f(x)]^2 \,dx \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi\int_{a}^{b}f(x) \,dx \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi\int_{b}^{a}[f(x)]^2 \,dx \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi[\int_{a}^{b}f(x) \,dx ]^2\)</span></label></p></div><div class="q-explanation"><p>The volume of a solid rotated around the <strong>x axis </strong>is <span class="math-tex">\(\large V=\pi\int_{a}^{b}[f(x)]^2 \,dx \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve <span class="math-tex">\(\large y = x^2\)</span></p><p>The region bounded by the curve and the lines x = 0 and y = 0 and y = a is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> y axis</strong>.</p><p>Which of the following represents the volume of this solid?</p><p><img alt="" src="../../files/integration/volume/quiz/q2.png" style="width: 450px; height: 334px;"></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large V=\pi\int_{0}^{a}y \,dy \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi\int_{0}^{a}x^2 \,dx \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi\int_{b}^{a}y^2 \,dx \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi\int_{a}^{0}x^2 \,dy \)</span></label></p></div><div class="q-explanation"><p>The volume of a solid rotated around the <strong>y axis </strong>is <span class="math-tex">\(\large V=\pi\int_{a}^{b}x^2 \,dy \)</span></p><p>Since y = x², then <span class="math-tex">\(\large V=\pi\int_{0}^{a}y \,dy \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve <span class="math-tex">\(\large f(x) = 2x(1-x)\)</span> and <span class="math-tex">\(\large g(x) = x(1-x)\)</span></p><p>The shaded region is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> x axis</strong>.</p><p>Which of the following represents the volume of this solid?</p><p><img alt="" src="../../files/integration/volume/quiz/q3.png" style="width: 450px; height: 334px;"></p><p> </p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large V=\pi\int_{0}^{1}[2x(1-x)]^2 \,dx -\pi\int_{0}^{1}[x(1-x)]^2 \,dx \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi\int_{1}^{0}[2x(1-x)]^2 -[x(1-x)]^2 \,dx \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi\int_{0}^{1}[2x(1-x) -x(1-x)]^2 \,dx \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large V=\pi\int_{0}^{1}[x(1-x)]^2 \,dx -\pi\int_{0}^{1}[2x(1-x)]^2 \,dx \)</span></label></p></div><div class="q-explanation"><p>To find the volume of a solid generated from the region between two curves, you find the volume generated from the "upper" curve and subtract the volume generated from the "lower" curve.</p><p>In this case, the volume generated from the "upper" curve is <span class="math-tex">\(\large V_{upper}=\pi\int_{0}^{1}[2x(1-x)]^2 \,dx\)</span></p><p>and the volume generated from the "lower" curve is <span class="math-tex">\(\large V_{lower}=\pi\int_{0}^{1}[x(1-x)]^2 \,dx\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The region bounded by the line y= x and the lines y = 0 and x = 1 and x = 2 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> x axis</strong>.</p><p>Which of the following is the volume of this solid?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{3\pi}{2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large3\pi\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large\frac{7\pi}{3}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large7\pi\)</span></label></p></div><div class="q-explanation"><p><img alt="" src="../../files/integration/volume/quiz/q4.png" style="width: 450px; height: 334px;"></p><p><span class="math-tex">\(\large V=\pi\int_{1}^{2}x^2 \,dx \\ \large V=\pi[\frac{x^3}{3}]_{1}^{2} \\ \large V=\pi([\frac{2^3}{3}]-[\frac{1^3}{3}]) \\ \large V=\pi([\frac{8}{3}]-[\frac{1^3}{3}]) \\ \large V=\frac{7}{3}\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve <span class="math-tex">\(\large f(x)=\sqrt{x}\)</span></p><p>The region bounded by the curve and the lines y = 0 and x = 0 and x = 2 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> x axis</strong>.</p><p>If the volume is <span class="math-tex">\(\large a\pi\)</span> , find the value of <strong><em>a</em></strong></p><p><strong><em><img alt="" src="../../files/integration/volume/quiz/q5.png" style="width: 400px; height: 342px;"></em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\pi\int_{0}^{2}(\sqrt{x})^2 \,dx \\ \large V=\pi\int_{0}^{2}x \,dx \\ \large V=\pi[\frac{x^2}{2}]_{0}^{2} \\ \large V=\pi([\frac{2^2}{2}]-[\frac{0^2}{2}]) \\ \large V=2\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve <span class="math-tex">\(\large f(x)=\frac{1}{x}\)</span></p><p>The region bounded by the curve and the lines y = 0 and x = 1 and x = 4 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> x axis</strong>.</p><p>If the volume is <span class="math-tex">\(\large a\pi\)</span> , find the value of <strong><em>a</em></strong></p><p><em>Give your answer as a decimal</em></p><p><strong><em><img alt="" src="../../files/integration/volume/quiz/q6.png" style="width: 400px; height: 342px;"></em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0.75"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\pi\int_{1}^{4}(\frac{1}{x})^2 \,dx \\ \large V=\pi\int_{1}^{4}x^{-2} \,dx \\ \large V=\pi[\frac{x^{-1}}{-1}]_{1}^{4} \\ \large V=\pi[\frac{-1}{x}]_{1}^{4} \\ \large V=\pi([\frac{-1}{4}]-[\frac{-1}{1}]) \\ \large V=\frac{3}{4}\pi\\ \large V=0.75\pi\\\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve <span class="math-tex">\(\large y = \frac{1}{x^2}\)</span></p><p>The region bounded by the curve and the lines x = 0 and y = 1 and y = 2 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> y axis</strong>.</p><p>Which of the following represents the volume of this solid?</p><p><img alt="" src="../../files/integration/volume/quiz/q7_1.png" style="width: 400px; height: 342px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{7\pi}{24}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large(\ln 2)\pi\\\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{\pi}{2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2\pi(\sqrt{2}-1)\)</span></label></p></div><div class="q-explanation"><p>The volume of a solid rotated around the <strong>y axis </strong>is <span class="math-tex">\(\large V=\pi\int_{a}^{b}x^2 \,dy \)</span></p><p>Since <span class="math-tex">\(\large y = \frac{1}{x^2}\)</span>, then <span class="math-tex">\(\large V=\pi\int_{1}^{2}\frac{1}{y} \,dy \)</span></p><p><span class="math-tex">\(\large V=\pi\int_{1}^{2}\frac{1}{y} \,dy \\ \large V=\pi[\ln |x|]_{1}^{2} \\ \large V=\pi(\ln 2-\ln 1)\\ \large V=(\ln 2)\pi\\\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve <span class="math-tex">\(\large f(x)=\frac{1}{\sqrt{x}}\)</span></p><p>The region bounded by the curve and the lines y = 0 and x = 1 and x = <span class="math-tex">\(\large e^3\)</span> is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> x axis</strong>.</p><p>If the volume is <span class="math-tex">\(\large a\pi\)</span> , find the value of <strong><em>a</em></strong></p><p><strong><em><img alt="" src="../../files/integration/volume/quiz/q8.png" style="width: 400px; height: 343px;"></em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\pi\int_{1}^{e^3}(\frac{1}{\sqrt{x}})^2 \,dx \\ \large V=\pi\int_{1}^{e^3}\frac{1}{x} \,dx \\ \large V=\pi[\ln| x|]_{1}^{e^3} \\ \large V=\pi([\ln e^3-\ln 1]) \\ \large V=\pi(\ln e^3)\\ \large V=3\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve <span class="math-tex">\(\large y = 2\ln x\)</span></p><p>The region bounded by the curve and the lines x = 0 and y = 0 and y = 1 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> y axis</strong>.</p><p>Which of the following represents the volume of this solid?</p><p><img alt="" src="../../files/integration/volume/quiz/q9.png" style="width: 400px; height: 343px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (2e^{0.5}-1)\pi\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large (e-1)\pi\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large (\frac{1}{2}e^{0.5}-1)\pi\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 8\pi\)</span></label></p></div><div class="q-explanation"><p>The volume of a solid rotated around the <strong>y axis </strong>is <span class="math-tex">\(\large V=\pi\int_{a}^{b}x^2 \,dy \)</span></p><p>Since</p><p style="margin-left: 40px;"> <span class="math-tex">\(\large y = 2\ln x\\ \large \frac{y}{2}=\ln x\\ \large e^{\frac{y}{2}}=x\\ \large (e^{\frac{y}{2}})^2=x^2\\ \large e^y=x^2\)</span>,</p><p>then</p><p><span class="math-tex">\(\large V=\pi\int_{0}^{1}e^y \,dy \)</span></p><p><span class="math-tex">\(\large V=\pi[e^y]_{0}^{1} \\ \large V=\pi(e^1-e^0)\\ \large V=(e-1)\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the curve <span class="math-tex">\(\large y=\sin x\)</span></p><p>The region bounded by the curve and the lines y = 0 and x = 0 and x = <span class="math-tex">\(\frac{\pi}{2}\)</span> is rotated <span class="math-tex">\(\large 2\pi\)</span> around the<strong> x axis</strong>.</p><p>If the volume is <span class="math-tex">\(\large \frac{\pi^2}{a}\)</span> , find the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/integration/volume/quiz/q10_1.png" style="width: 400px; height: 343px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\pi\int_{0}^{\frac{\pi}{2}}(\sin x)^2 \,dx\)</span></p><p>To integrate sin²x, we need to use the double angle formula for cos2x</p><p><span class="math-tex">\(\large \cos2x\equiv1-2\sin^2x\\ \large 2\sin^2x\equiv 1-\cos2x\\ \large \sin^2x\equiv \frac{1}{2}-\frac{\cos2x}{2}\)</span></p><hr class="hidden-separator"><p><span class="math-tex">\(\large V=\pi\int_{0}^{\frac{\pi}{2}}(\sin x)^2 \,dx\\ \large V=\pi\int_{0}^{\frac{\pi}{2}}(\frac{1}{2}-\frac{\cos2x}{2}) \,dx\\ \large V=\pi[\frac{x}{2}-\frac{\sin2x}{4}]_{0}^{\frac{\pi}{2}} \\ \large V=\pi([\frac{\frac{\pi}{2}}{2}-\frac{\sin\pi}{4}]-[\frac{0}{2}-\frac{\sin0}{4}])\\ \large V=\pi(\frac{\pi}{4})\\ \large V=\frac{\pi^2}{4}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"></div> </div> <div class="panel panel-default panel-has-colored-body"> <div class="panel-heading"> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="938"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Show that <span class="math-tex">\(\large \int {x^2e^{2x}\,dx}=\frac{1}{4}e^{2x}(2x^2-2x+1)+c\)</span></p> <p>The following graph shows the function <span class="math-tex">\(\large f(x)=xe^x\)</span></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2_1.png" style="width: 400px; height: 401px;"></p> <p>b) Show that the equation of the tangent to the graph at x = 1 has the equation <span class="math-tex">\(\large y=(2e)x-e\)</span></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2_2.png" style="width: 400px; height: 356px;"></p> <p>The region bounded by <span class="math-tex">\(\large f\)</span>, the tangent <span class="math-tex">\(\large y=(2e)x-e\)</span> and y = 0 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the x axis</p> <p>c) Find the volume of this solid in terms of <span class="math-tex">\(\large \pi\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) this is an integration by parts question</p> <p><content>c) the volume of the solid can be found by subtracting the volume of a cone from the volume of the curved solid </content></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2_hint.png" style="width: 400px; height: 207px;"></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a)</p> <p><img alt="" src="../../files/integration/volume/esqs/esq2a_1.png" style="width: 600px; height: 219px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2a_2.png" style="width: 600px; height: 244px;"></p> <p>b)</p> <p><img alt="" src="../../files/integration/volume/esqs/esq2b.png" style="width: 600px; height: 361px;"></p> <p>c)</p> <p><img alt="" src="../../files/integration/volume/esqs/esq2c_1.png" style="width: 600px; height: 374px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2c_2.png" style="width: 600px; height: 143px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2c_3.png" style="width: 600px; height: 341px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq2c_4.png" style="width: 600px; height: 145px;"></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="937"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following graph shows the curve defined by the equation <span class="math-tex">\(\large (x-1)^2+y^2=4\)</span>.</p> <p>The region bounded by the curve and the lines y = 0 and x = 0 is rotated <span class="math-tex">\(\large 2\pi\)</span> around the x axis.</p> <p>Find the volume of this solid in terms of <span class="math-tex">\(\large \pi\)</span></p> <p><img alt="" src="../../files/integration/volume/esqs/esq1.png" style="width: 450px; height: 400px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>The volume of the solid is found using the formula</p> <p><content><span class="math-tex">\(\large V= \int_{0}^{3} y^2 \,dx \\ \large V= \int_{0}^{3} (4-(x-1)^2 \,)dx \)</span></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/integration/volume/esqs/esq1ans_1.png" style="width: 800px; height: 445px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="939"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The graph shows <span class="math-tex">\(\large f(x) = -\arctan(1-x^2)\)</span>, the tangent to the curve at (1 , 0) and the tangent to the curve at the point <span class="math-tex">\(\large(-\frac{\pi}{4},0)\)</span>.</p> <p>The shaded region is bounded by the curve and the two tangents. This region is rotated <span class="math-tex">\(\large 2\pi\)</span> around the <strong>y axis</strong> to forma solid.</p> <p>Find the volume of this solid correct to 3 significant figures.</p> <p><img alt="" src="../../files/integration/volume/esqs/esq3.png" style="width: 400px; height: 402px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>You can use your calculator for this question.</p> <p>Ensure that your calculator is in radian mode.</p> <p><content>You can find the volume gene</content><content>rated by rotating the trapezium around the y axis and subtract the volume generated by rotating the curve around the y axis</content></p> <p><img alt="" src="../../files/integration/volume/esqs/esq4_hint.png" style="width: 600px; height: 246px;"></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/integration/volume/esqs/esq3_a1.png" style="width: 600px; height: 184px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq3_a2.png" style="width: 600px; height: 371px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq3_a3.png" style="width: 600px; height: 386px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq3_a4.png" style="width: 600px; height: 90px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="940"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows the graph of <span class="math-tex">\(\large x^2=\cos^3y\)</span> for <span class="math-tex">\(\large -\frac{\pi}{2}\le y\le\frac{\pi}{2}\)</span></p> <p>The shaded region <strong>R </strong>is the area bounded by the curve, the y axis and the lines <span class="math-tex">\(\large y=-\frac{\pi}{2}\)</span> and <span class="math-tex">\(\large y=\frac{\pi}{2}\)</span>.</p> <p>The rgion is rotated about the <strong>y axis</strong> through <span class="math-tex">\(\large 2\pi\)</span> to form a solid.</p> <p>Show that the volume of the solid is <span class="math-tex">\(\large \frac{4\pi}{3}\)</span></p> <p><img alt="" src="../../files/integration/volume/esqs/esq4.png" style="width: 450px; height: 401px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>To perfom the integration, write</p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \cos^3y=\cos y\cdot\cos^2y=\cos y\ (1-\sin^2y)\)</span><content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/integration/volume/esqs/esq4a_1.png" style="width: 600px; height: 225px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq4a_2.png" style="width: 600px; height: 251px;"></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="941"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows the graph of the function <span class="math-tex">\(\large f(x)=\frac{\sqrt{x}}{\cos x}\)</span> for <span class="math-tex">\(\large 0\le x \le\frac{\pi}{2}\)</span></p> <p>The shaded region is the area bounded by <strong><em>f</em></strong>, the x axis, x= 0 and x = <span class="math-tex">\(\large \frac{\pi}{3}\)</span></p> <p>The region is rotated about the x axis through <span class="math-tex">\(\large 2\pi\)</span> to form a solid.</p> <p>Find the volume of the solid.</p> <p><img alt="" src="../../files/integration/volume/esqs/esq5.png" style="width: 450px; height: 475px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>It is the integration that makes this question difficult.</p> <p><content>You will need to use both integration by parts and integration by substitution/recognition.</content></p> <p>For the integration by substitution, let</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large u=x \qquad \frac{dv}{dx}=\sec^2x\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/integration/volume/esqs/esq5a_1.png" style="width: 600px; height: 333px;"></p> <p><img alt="" src="../../files/integration/volume/esqs/esq5a_2.png" style="width: 600px; height: 273px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="644"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Volume of Revolution</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div 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