A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The diagram below shows a sketch of part of the curve $y=\mathrm{f}(x)$. The area of the shaded region under the curve between $x=2$ and $x=6$ is $A$ units$^2$. This shaded region is split into 6 vertical strips of equal width, $h$.</p>
<div class="graph-block" data-graph-config="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" data-width="360" data-height="265" style="width:360px;height:265px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0" data-list-item-path="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0"><ol class="i-list"><li data-list-item-id="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0-sub-0" data-list-item-path="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0/h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0-sub-0"><div class="list-item-paragraph">State the value of $h$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0-sub-1" data-list-item-path="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0/h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0-sub-1"><div class="list-item-paragraph">Using a sketch, explain whether $\displaystyle\sum_{n=0}^{5} hf(2+hn)$ is an overestimate or underestimate of $A$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0-sub-2" data-list-item-path="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0/h2-2025-hci-promo-q7-hci-q7-item-block-2-item-0-sub-2"><div class="list-item-paragraph"><p>Comment on how a better estimate in part <strong>(a)(ii)</strong> could be found.</p><span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-1" data-list-item-path="h2-2025-hci-promo-q7-hci-q7-item-block-2-item-1"><div class="list-item-paragraph">The region bounded by the curve $y=\sqrt{\dfrac{2}{x\ln x^2}}$, where&nbsp;$x&gt;1$, the $x$-axis and the lines $x=2$ and $x=6$, is rotated through $2\pi$ radians about the $x$-axis. Using calculus, find the exact volume of the solid obtained, simplifying your answer.<span class="inline-marks">[3]</span></div></li></ol>

The diagram below shows a sketch of part of the curve $y=\mathrm{f}(x)$. The area of the shaded region under the curve between $x=2$ and $x=6$ is $A$ units$^2$. This shaded region is split into 6 vertical strips of equal width, $h$.

The region bounded by the curve $y=\sqrt{\dfrac{2}{x\ln x^2}}$, where $x>1$, the $x$-axis and the lines $x=2$ and $x=6$, is rotated through $2\pi$ radians about the $x$-axis. Using calculus, find the exact volume of the solid obtained, simplifying your answer.

Volume of Revolution about the x-axis

Riemann Sums

2025 HCI Promo Q7

Applications of Integration

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.<span class="inline-marks">[2]</span></div></li></ol>
<p>A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>Using the result in part <strong>(a)</strong>, find the exact volume generated when&nbsp;$R$ is rotated $360^\circ$ about the $x$-axis.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Find the volume generated when&nbsp;$R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.<span class="inline-marks">[3]</span></div></li></ol>

Show that $\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.

A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$.

Find the volume generated when $R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.

Derivatives of Natural Logarithmic Functions

Volume of Revolution about the y-axis

Product Rule

N2025 P1 Q7

Techniques of Differentiation

<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><ol class="i-list"><li data-list-item-id="imported-item-block-1-item-0-sub-0" data-list-item-path="imported-item-block-1-item-0/imported-item-block-1-item-0-sub-0"><div class="list-item-paragraph">Find $\int x^2 \cos n x \,\mathrm{d}x$, where&nbsp;$n$ is a positive integer.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-0-sub-1" data-list-item-path="imported-item-block-1-item-0/imported-item-block-1-item-0-sub-1"><div class="list-item-paragraph">Hence find $\int_{\pi}^{2\pi} x^2 \cos n x \,\mathrm{d}x$, giving your answers in the form $a\frac{\pi}{n^2}$, where the possible values of $a$ are to be determined.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">The region bounded by the curve $y = \frac{x\sqrt{x}}{9-x^2}$, the $x$-axis and the lines $x = 0$ and $x = 2$ is rotated through $2\pi$ radians about the $x$-axis. Use the substitution $u = 9 - x^2$ to find the exact volume of the solid obtained, simplifying your answer.<span class="inline-marks">[5]</span></div></li></ol>

The region bounded by the curve $y = \frac{x\sqrt{x}}{9-x^2}$, the $x$-axis and the lines $x = 0$ and $x = 2$ is rotated through $2\pi$ radians about the $x$-axis. Use the substitution $u = 9 - x^2$ to find the exact volume of the solid obtained, simplifying your answer.

Integration by Parts

Integration by Substitution

N2016 P2 Q2 

Techniques of Integration

<p>The curve $C_1$ has equation $y = \frac{x^2 - 8x + 28}{2 - x}$.</p>
<p>The curve $C_2$ has equation $\frac{(x-6)^2}{16} + \frac{y^2}{25} = 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-8-a" data-list-item-path="item-8-a"><div class="list-item-paragraph">Sketch $C_1$ and $C_2$ on the same diagram, stating the coordinates of any vertices, turning points, intersections with the $x$-axis and the equations of any asymptotes.<span class="inline-marks">[7]</span></div></li><li data-list-item-id="item-8-b" data-list-item-path="item-8-b"><div class="list-item-paragraph">Find the volume of solid obtained when the smaller region bounded by $C_1$ and $C_2$ is rotated through $2\pi$ radians about the $x$-axis.<span class="inline-marks">[4]</span></div></li></ol>

The curve $C_1$ has equation $y = \frac{x^2 - 8x + 28}{2 - x}$.

The curve $C_2$ has equation $\frac{(x-6)^2}{16} + \frac{y^2}{25} = 1$.

Sketch $C_1$ and $C_2$ on the same diagram, stating the coordinates of any vertices, turning points, intersections with the $x$-axis and the equations of any asymptotes.

Find the volume of solid obtained when the smaller region bounded by $C_1$ and $C_2$ is rotated through $2\pi$ radians about the $x$-axis.

Sketching Rational Functions

Asymptotes and Discontinuities

Graphs of Conic Sections

2025 EJC P2 Q3 

Graphing Techniques

<ol class="abc-list"><li data-list-item-id="item-a-3" data-list-item-path="item-a-3"><ol class="i-list"><li data-list-item-id="item-a-3-i" data-list-item-path="item-a-3/item-a-3-i"><div class="list-item-paragraph"><p>Show that&nbsp;$\frac{3x^2}{(x+1)(3x^2+x+1)}$ can be expressed as<br></p></div><div class="list-item-paragraph"><p style="text-align: center;">$\frac{A}{x+1} + \frac{Bx+C}{3x^2+x+1}$,<br></p></div><div class="list-item-paragraph"><p>where&nbsp;$A$, $B$ and $C$ are constants to be determined.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-3-ii" data-list-item-path="item-a-3/item-a-3-ii"><div class="list-item-paragraph">Hence, find $\displaystyle \displaystyle\int \frac{3x^2}{(x+1)(3x^2+x+1)} \, \mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="item-b-3" data-list-item-path="item-b-3"><div class="list-item-paragraph">The region $R$ is bounded by the curve $y = \frac{3x^2}{(x+1)(3x^2+x+1)}$, the line $x=4$ and the $x$-axis.</div><ol class="i-list"><li data-list-item-id="item-b-3-i" data-list-item-path="item-b-3/item-b-3-i"><div class="list-item-paragraph">Sketch the graph of $y = \frac{3x^2}{(x+1)(3x^2+x+1)}$ for&nbsp;$x \ge 0$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region $R$ on your sketch.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-3-ii" data-list-item-path="item-b-3/item-b-3-ii"><div class="list-item-paragraph">Find the exact area of $R$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-3-iii" data-list-item-path="item-b-3/item-b-3-iii"><div class="list-item-paragraph">Find the volume of the solid generated when&nbsp;$R$ is rotated through $2\pi$ radians about the $x$-axis.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Integration by Splitting the Numerator

Integration Using Partial Fractions

Area Bounded by Curves

Partial Fractions

Integration by Completing the Square

Standard Integrals with Quadratic Denominators

2025 CJC P1 Q11

<p>The parametric equations of the curve $C$ are</p>
<p style="text-align: center">$x=\tan \theta -1$, $y=2\sec \theta $, where&nbsp;$-\frac{\pi }{2}&lt;\theta &lt;\frac{\pi }{2}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760961275279-njpvri7j9" data-list-item-path="item-1760961275279-njpvri7j9"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\theta $, leaving your answer in a single trigonometric function.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760961275936-uic3edge1" data-list-item-path="item-1760961275936-uic3edge1"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point where it cuts the $y$-axis, leaving your answer in exact form.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760961276913-61m5k7kii" data-list-item-path="item-1760961276913-61m5k7kii"><div class="list-item-paragraph">Find the Cartesian equation of $C$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760961277949-yh1yw236v" data-list-item-path="item-1760961277949-yh1yw236v"><div class="list-item-paragraph">Find the equations of the asymptotes and state the coordinates of any turning point(s).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760961279831-43bl3h9nr" data-list-item-path="item-1760961279831-43bl3h9nr"><div class="list-item-paragraph">The region bounded by the tangent found in part <strong>(b)</strong>, the curve $C$ and the line $x=-1$ is rotated completely about the $x$-axis. Calculate the volume obtained.<span class="inline-marks">[3]</span></div></li></ol>

The parametric equations of the curve $C$ are

$x=\tan \theta -1$, $y=2\sec \theta $, where $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$.

Tangents and Normals to Parametric Curves

Derivatives of Reciprocal Trigonometric Functions

Differentiation of Parametric Equations

Converting Parametric Equations to Cartesian Form

2024 HCI Promo Q12

Applications of Differentiation

<ol class="abc-list"><li data-list-item-id="item-1760955635047-9fe3nmrk6" data-list-item-path="item-1760955635047-9fe3nmrk6"><div class="list-item-paragraph">Using the substitution $u=\sqrt{4x+1}$, show that$\displaystyle\displaystyle \displaystyle\int{\frac{\sqrt{4x+1}}{x-2}}\mathrm{ d}x=\displaystyle\displaystyle \displaystyle\int{\frac{2{{u}^{2}}}{{{u}^{2}}-9}}\mathrm{ d}u$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760955635537-h5mrbedmj" data-list-item-path="item-1760955635537-h5mrbedmj"><div class="list-item-paragraph">The region $R$ lies in the first quadrant and is bounded by the curve $y=\frac{\sqrt{4x+1}}{x-2}$ and the lines $x=6$, $x=12$ and $y=0.5$.</div><ol class="i-list"><li data-list-item-id="subitem-1760955652450-22oklxjp5" data-list-item-path="item-1760955635537-h5mrbedmj/subitem-1760955652450-22oklxjp5"><div class="list-item-paragraph"><p>Sketch the graph of $y=\frac{\sqrt{4x+1}}{x-2}$ for&nbsp;$x \geqslant 0$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region $R$ on your sketch.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760955653573-0vzpnih3e" data-list-item-path="item-1760955635537-h5mrbedmj/subitem-1760955653573-0vzpnih3e"><div class="list-item-paragraph">Find the exact area of $R$, giving your answer in the form $a+b\ln c$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="subitem-1760955654465-kpwamcczy" data-list-item-path="item-1760955635537-h5mrbedmj/subitem-1760955654465-kpwamcczy"><div class="list-item-paragraph"><p>Find the volume of the solid generated when&nbsp;$R$ is rotated through $2\pi $ radians about the $x$- axis.</p><p>Give your answer correct to $2$ decimal places.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>

Integrals Involving Long Division

Area bounded by 2 curves

N2024 P1 Q10

<p>The region bounded by $y=x\sqrt{\sin \left( x \right)}$, the $x$-axis, and the lines $x=0$ and $x=\pi $ is rotated through $2\pi $ radians about the $x$-axis. Find the exact volume of the solid generated.<span class="inline-marks">[5]</span></p>

The region bounded by $y=x\sqrt{\sin \left( x \right)}$, the $x$-axis, and the lines $x=0$ and $x=\pi $ is rotated through $2\pi $ radians about the $x$-axis. Find the exact volume of the solid generated.

2024 NJC Promo Q2

<p>If $k$ is a non-zero integer, find $\displaystyle\int_{-\pi }^{\pi }{\sin kx\,\mathrm{d}x}$ and $\displaystyle\int_{-\pi }^{\pi }{\cos kx\,\mathrm{d}x}$.</p><p style="text-align: right;">[2]</p>
<p>Let $m$ and $n$ be positive integers.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{-\pi }^{\pi }{\sin \left( mx \right)\sin \left( nx \right)\mathrm{d}x=\left\{ \begin{aligned}
 \pi \mathrm{  if  }m&=n, \\
 0\mathrm{  if  }\,m&\ne n. \\
\end{aligned} \right.}$</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">By considering the formula for $\cos \left( mx+nx \right)$, evaluate</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{-\pi }^{\pi }{\cos \left( mx \right)\cos \left( nx \right)\mathrm{d}x}$.</div></li></ol>
<p>Suppose $a$ is a positive integer.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that $\displaystyle\int_{-\pi }^{\pi }{\left| \sin \left( ax \right) \right|\mathrm{d}x}=4$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence, find the exact volume of the solid formed when the region bounded by the curve $y=\left| \sin \left( ax \right) \right|+1$, the $x$-axis and the lines $x=-\pi $ and $x=\pi $, is rotated completely about the $x$-axis.</div><p style="text-align: right;">[3]</p></li></ol>

If $k$ is a non-zero integer, find $\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\sin kx\,\mathrm{d}x}$ and $\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\cos kx\,\mathrm{d}x}$.

Integrals involving Modulus Functions

Integrating Powers of Sine and Cosine

Integrable Forms of Trigonometric Functions

Trigonometric Integration Using Product-to-Sum Formulae

2019 NJC P2 Q4

<p>The diagram shows the region ${R}$, which is bounded by the curve ${(y-2)^{2}=2x-3}$ and the line ${x=3.5}$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the</p><ol class="i-list"><li data-list-item-id="item-1790411500021-19e5lk948" data-list-item-path="item-1790411500021-19e5lk948"><div class="list-item-paragraph"><p>area of region ${R}$.</p></div></li><li data-list-item-id="item-1790411500793-i7g0t6tg3" data-list-item-path="item-1790411500793-i7g0t6tg3"><div class="list-item-paragraph"><p>volume when&nbsp;${R}$ is rotated through four right angles about the ${x}$-axis.</p></div></li><li data-list-item-id="item-1790411501589-99mij59u7" data-list-item-path="item-1790411501589-99mij59u7"><div class="list-item-paragraph"><p>volume when&nbsp;${R}$ is rotated through ${2\pi}$ about the ${y}$-axis.</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="256" data-height="223" data-proportional style="width:256px;height:223px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:256px;--graph-aspect:256 / 223;"></div></div></div></div>

The diagram shows the region ${R}$, which is bounded by the curve ${(y-2)^{2}=2x-3}$ and the line ${x=3.5}$.

Volume of Revolution: Roots of Equations

<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><p>Find the volumes of solid formed when region ${R}$ is rotated through ${2\pi}$ about</p><ol class="abc-list"><li data-list-item-id="item-1790409768941-4f23ijx0s" data-list-item-path="item-1790409768941-4f23ijx0s"><div class="list-item-paragraph"><p>${y}$-axis,</p></div></li><li data-list-item-id="item-1790409769593-b62yblex9" data-list-item-path="item-1790409769593-b62yblex9"><div class="list-item-paragraph"><p>${x}$-axis.</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="244" data-height="187" data-proportional style="width:244px;height:187px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:244px;--graph-aspect:244 / 187;"></div></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the volumes of solid formed when region ${R}$ is rotated through ${2\pi}$ about</p><ol class="abc-list"><li data-list-item-id="item-1790405400625-njfim6qjc" data-list-item-path="item-1790405400625-njfim6qjc"><div class="list-item-paragraph"><p>${y}$-axis,</p></div></li><li data-list-item-id="item-1790405401525-9ne1comxv" data-list-item-path="item-1790405401525-9ne1comxv"><div class="list-item-paragraph"><p>${x}$-axis.</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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data-width="351" data-height="333" data-proportional style="width:351px;height:333px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:351px;--graph-aspect:351 / 333;"></div></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the volumes of solid formed when region ${R}$ is rotated through ${2\pi}$ about</p><ol class="abc-list"><li data-list-item-id="item-1790411074084-ml5uwwzi8" data-list-item-path="item-1790411074084-ml5uwwzi8"><div class="list-item-paragraph"><p>${x}$-axis,</p></div></li><li data-list-item-id="item-1790411074774-22zuzkks6" data-list-item-path="item-1790411074774-22zuzkks6"><div class="list-item-paragraph"><p>${y}$-axis.</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="334" data-height="307" data-proportional style="width:334px;height:307px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:334px;--graph-aspect:334 / 307;"></div></div></div></div>

<p>A curve ${C}$ has equation ${y = \frac{3 - x}{x - 1}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch ${C}$, stating the equations of the asymptotes.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the exact volume of the solid generated when the region bounded by ${C}$, the ${x}$-axis and the line ${x=9}$ is rotated through ${2\pi}$ radians about the ${x}$-axis. Leave your answer in the form ${\pi (a + b \ln 2)}$, where ${a}$ and ${b}$ are constants to be determined.</div><p style="text-align: right;">[5]</p></li></ol>

A curve ${C}$ has equation ${y = \frac{3 - x}{x - 1}}$.

2024 TJC P2 Q1

<p>Find the total volume of solids formed when regions $P$ and $Q$ are rotated through $2\pi $ about $x$-axis.</p>
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Find the total volume of solids formed when regions $P$ and $Q$ are rotated through $2\pi $ about $x$-axis.

Volume of Revolution

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the volumes of solid formed when region $R$ is rotated through $2\pi $ about</p><ol class="abc-list"><li data-list-item-id="item-1790400866275-lzq08pgwt" data-list-item-path="item-1790400866275-lzq08pgwt"><div class="list-item-paragraph"><p>$y$-axis,</p></div></li><li data-list-item-id="item-1790400867129-xfndc7i5s" data-list-item-path="item-1790400867129-xfndc7i5s"><div class="list-item-paragraph"><p>$x$-axis,</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="278" data-height="246" data-proportional style="width:278px;height:246px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:278px;--graph-aspect:278 / 246;"></div></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><p>Find the volumes of solid formed when region $R$ is rotated through $2\pi $ about</p><ol class="abc-list"><li data-list-item-id="item-1790400458572-bja0be2ze" data-list-item-path="item-1790400458572-bja0be2ze"><div class="list-item-paragraph"><p>$y$-axis,</p></div></li><li data-list-item-id="item-1790400459696-k2kdjmpha" data-list-item-path="item-1790400459696-k2kdjmpha"><div class="list-item-paragraph"><p>$x$-axis,</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="244" data-height="241" data-proportional style="width:244px;height:241px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:244px;--graph-aspect:244 / 241;"></div></div></div></div>

<p>Find the volumes of solid formed when region $R$ is rotated through $2\pi $ about</p>
<ol class="abc-list"><li data-list-item-id="item-1760758152540-uhcpq8kt7" data-list-item-path="item-1760758152540-uhcpq8kt7"><div class="list-item-paragraph">$y$-axis,</div></li><li data-list-item-id="item-1760758153042-beywfvtim" data-list-item-path="item-1760758153042-beywfvtim"><div class="list-item-paragraph">$x$-axis,</div></li></ol>
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Find the volumes of solid formed when region $R$ is rotated through $2\pi $ about

<p>Find the volume of solid formed when region $R$ is rotated through $2\pi $ about $x$-axis.</p>
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Find the volume of solid formed when region $R$ is rotated through $2\pi $ about $x$-axis.

<p>Find the volume of solid formed when region $R$ is rotated through $2\pi $ about $x$-axis.</p>
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