Volumes of Revolution

3 The curve \(y ^ { 2 } = x - 1\) for \(1 \leq x \leq 3\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed.

2 The function f is defined by $$f ( x ) = 2 x + 3 \quad 0 \leq x \leq 5$$ The region \(R\) is enclosed by \(y = \mathrm { f } ( x ) , x = 5\), the \(x\)-axis and the \(y\)-axis.
The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Give an expression for the volume of the solid formed.
Tick ( ✓ ) one box. \(\pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) d x\) \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-02_113_108_1539_1000} \(\pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) ^ { 2 } \mathrm {~d} x\) \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-02_115_108_1699_1000} \(2 \pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) d x\) □ \(2 \pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) ^ { 2 } \mathrm {~d} x\) □

4 The equation of a curve is \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm {~K} ^ { 2 } + \mathrm { x } ^ { 2 } } }\), where \(k\) is a positive constant. The region between the \(x\)-axis, the \(y\)-axis and the line \(x = k\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Given that the volume of the solid of revolution formed is 1 unit \({ } ^ { 3 }\), find the exact value of \(k\).

8. A curve has equation \(y = 2 + \mathrm { e } ^ { \frac { 1 } { 2 } x }\). The region \(R\) is bounded by the curve and by the straight lines \(x = 0 , x = 4\) and \(y = 0\). Find the exact volume of the solid obtained when \(R\) is rotated completely about the x-axis.
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5. \begin{figure}[h] \end{figure} Diagram not drawn to scale The finite region \(S\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = \ln 4\) and the curve with equation $$y = \mathrm { e } ^ { x } + 2 \mathrm { e } ^ { - x } , \quad x \geqslant 0$$ The region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Use integration to find the exact value of the volume of the solid generated. Give your answer in its simplest form.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

12. \includegraphics[max width=\textwidth, alt={}, center]{368f5263-8f2e-40a9-8bc0-f074d8a98ee1-26_689_1203_182_447} The figure shows part of the graph of \(y = ( x - 3 ) \sqrt { \ln x }\). The portion of the graph below the \(x\)-axis is rotated by \(2 \pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\). Determine the exact volume of \(S\).
[0pt] [BLANK PAGE]

1 \includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-2_668_554_260_797} The diagram shows part of the curve \(y = x ^ { 2 } + 1\). Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

3 Fig. 3 shows the curve \(y = x ^ { 4 }\) and the line \(y = 4\). \begin{figure}[h] \end{figure} The finite region enclosed by the curve and the line is rotated through \(180 ^ { \circ }\) about the \(y\)-axis. Find the exact volume of revolution generated.

16 The function f is defined by $$f ( x ) = \frac { a x + 5 } { x + b }$$ where \(a\) and \(b\) are constants. The graph of \(y = \mathrm { f } ( x )\) has asymptotes \(x = - 2\) and \(y = 3\) 16

1. Write down the value of \(a\) and the value of \(b\) 16
2. The diagram shows the graph of \(y = \mathrm { f } ( x )\) and its asymptotes.
   The shaded region \(R\) is enclosed by the graph of \(y = \mathrm { f } ( x )\), the \(x\)-axis and the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-20_858_1002_1267_504} 16
3. 1. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. 16
4. (ii) The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis to form a solid.
   Find the volume of this solid.
   Give your answer to three significant figures.
   [0pt] [4 marks]

1. The curve \(C\) has equation \(y = 2 x ^ { 3 } , 0 \leqslant x \leqslant 2\).

The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Using calculus, find the area of the surface generated, giving your answer to 3 significant figures.

7. A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x ^ { 2 } + y ^ { 2 } = r ^ { 2 }\) where \(r\) is a constant.

1. Show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \frac { r ^ { 2 } } { r ^ { 2 } - x ^ { 2 } }\)
2. Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4 \pi r ^ { 2 }\).
3. Write down the length of the arc of the curve \(y = \sqrt { } \left( 1 - x ^ { 2 } \right)\) from \(x = 0\) to \(x = 1\)

3 A curve has equation \(y = \mathrm { e } ^ { x }\) for \(\ln \frac { 4 } { 3 } \leqslant x \leqslant \ln \frac { 12 } { 5 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).

1. Use the substitution \(u = \mathrm { e } ^ { x }\) to show that $$A = 2 \pi \int _ { \frac { 4 } { 3 } } ^ { \frac { 12 } { 5 } } \sqrt { 1 + u ^ { 2 } } \mathrm {~d} u$$
2. Use the substitution \(u = \sinh v\) to show that $$A = \pi \left( \frac { 904 } { 225 } + \ln \frac { 5 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-06_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-07_2726_35_97_20}

10 The equation of a curve is \(y = \frac { 4 } { 2 x - 1 }\).

1. Find, showing all necessary working, the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Given that the line \(2 y = x + c\) is a normal to the curve, find the possible values of the constant \(c\).

9 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-4_719_670_264_735} The diagram shows the curve \(y = \sqrt { } ( 3 x + 1 )\) and the points \(P ( 0,1 )\) and \(Q ( 1,2 )\) on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 2\).

1. Find the area of the shaded region.
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Tangents are drawn to the curve at the points \(P\) and \(Q\).
3. Find the acute angle, in degrees correct to 1 decimal place, between the two tangents.

10 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-14_631_689_274_721} The diagram shows part of the curve with equation \(y = \frac { 4 } { ( 2 x - 1 ) ^ { 2 } }\) and parts of the lines \(x = 1\) and \(y = 1\). The curve passes through the points \(A ( 1,4 )\) and \(B , \left( \frac { 3 } { 2 } , 1 \right)\).

1. Find the exact volume generated when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. A triangle is formed from the tangent to the curve at \(B\), the normal to the curve at \(B\) and the \(x\)-axis. Find the area of this triangle.

3 Find the exact volume generated when the region enclosed between the \(x\)-axis and the portion of the curve \(y = \sin x\) between \(x = 0\) and \(x = \pi\) is rotated completely about the \(x\)-axis.

3 Find the exact volume generated when the region enclosed between the \(x\)-axis and the portion of the curve \(y = \sin x\) between \(x = 0\) and \(x = \pi\) is rotated completely about the \(x\)-axis.

1. Find the value of \(p\), the value of \(m\) and the value of \(n\).
2. Show that the equation of \(C\) can be written in the form \(y = r + \mathrm { f } ( x - h )\) and specify the function f and the constants \(r\) and \(h\). The region bounded by \(C\), the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 3 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Find the volume of the solid formed.

6

1. Use integration by parts to find \(\int \frac { \ln ( 3 x ) } { x ^ { 2 } } \mathrm {~d} x\).
2. The region bounded by the curve \(y = \frac { \ln ( 3 x ) } { x }\), the \(x\)-axis from \(\frac { 1 } { 3 }\) to 1 , and the line \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid. Find the exact value of the volume of the solid generated.
   [0pt] [7 marks]

4. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation $$y = \sqrt { } \left( \frac { 2 x } { 3 x ^ { 2 } + 4 } \right) , x \geqslant 0$$ The finite region \(S\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 2\) The region \(S\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis.
Use integration to find the exact value of the volume of the solid generated, giving your answer in the form \(k \ln a\), where \(k\) and \(a\) are constants.

9. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 3, has equation $$y = \frac { x ^ { - \frac { 1 } { 4 } } } { \sqrt { 1 + x } ( \arctan \sqrt { x } ) }$$ The region \(R\), shown shaded in Figure 3, is bounded by \(C\), the line with equation \(x = 3\), the \(x\)-axis and the line with equation \(x = \frac { 1 } { 3 }\) The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid.
Using the substitution \(\tan u = \sqrt { x }\)

1. show that the volume \(V\) of the solid formed is given by $$k \int _ { a } ^ { b } \frac { 1 } { u ^ { 2 } } \mathrm {~d} u$$ where \(k , a\) and \(b\) are constants to be found.
2. Hence, using algebraic integration, find the value of \(V\) in simplest form.

9 The shape of a vase can be modelled by rotating the curve with equation \(16 x ^ { 2 } - ( y - 8 ) ^ { 2 } = 32\) between \(y = 0\) and \(y = 16\) completely about the \(\boldsymbol { y }\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_890_1210_1555_424} The vase has a base.
Find the volume of water needed to fill the vase, giving your answer as an exact value.

9

1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
   \end{figure} Using this model,
   (A) find the greatest height of the tunnel,
   (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
   Hence estimate the volume of earth removed when the tunnel is re-shaped.

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central cross-section \(A O B C D\) of a circular bird bath, which is made of concrete. Measurements of the height and diameter of the bird bath, and the depth of the bowl of the bird bath have been taken in order to estimate the amount of concrete that was required to make this bird bath. Using these measurements, the cross-sectional curve CD, shown in Figure 2, is modelled as a curve with equation $$y = 1 + k x ^ { 2 } \quad - 0.2 \leqslant x \leqslant 0.2$$ where \(k\) is a constant and where \(O\) is the fixed origin.
The height of the bird bath measured 1.16 m and the diameter, \(A B\), of the base of the bird bath measured 0.40 m , as shown in Figure 1.

1. Suggest the maximum depth of the bird bath.
2. Find the value of \(k\).
3. Hence find the volume of concrete that was required to make the bird bath according to this model. Give your answer, in \(\mathrm { m } ^ { 3 }\), correct to 3 significant figures.
4. State a limitation of the model. It was later discovered that the volume of concrete used to make the bird bath was \(0.127 \mathrm {~m} ^ { 3 }\) correct to 3 significant figures.
5. Using this information and the answer to part (c), evaluate the model, explaining your reasoning.

8 The equation of a curve is \(y = \sqrt { } \left( 8 x - x ^ { 2 } \right)\). Find

1. an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and the coordinates of the stationary point on the curve,
2. the volume obtained when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

9 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-4_602_899_248_625} The diagram shows part of the curve \(y = x + \frac { 4 } { x }\) which has a minimum point at \(M\). The line \(y = 5\) intersects the curve at the points \(A\) and \(B\).

1. Find the coordinates of \(A , B\) and \(M\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

9 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-4_547_1401_264_370} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \sqrt { } ( 1 + 2 x )\) and its maximum point \(M\). The shaded region between the curve and the axes is denoted by \(R\).

1. Find the \(x\)-coordinate of \(M\).
2. Find by integration the volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.

1. Find the coordinates of \(A\).
2. Find the volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in the form \(\frac { \pi } { a } ( b \sqrt { c } - d )\), where \(a , b , c\) and \(d\) are integers.
3. Find an exact expression for the perimeter of the shaded region.

10 The function f is defined by \(\mathrm { f } ( x ) = 2 x + ( x + 1 ) ^ { - 2 }\) for \(x > - 1\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\) and hence verify that the function f has a minimum value at \(x = 0\). \includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-4_515_920_959_609} The points \(A \left( - \frac { 1 } { 2 } , 3 \right)\) and \(B \left( 1,2 \frac { 1 } { 4 } \right)\) lie on the curve \(y = 2 x + ( x + 1 ) ^ { - 2 }\), as shown in the diagram.
2. Find the distance \(A B\).
3. Find, showing all necessary working, the area of the shaded region. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
   To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

9. \begin{figure}[h] \end{figure} Figure 1 is a sketch of the curve \(C\) with equation $$y = 2 x ^ { \frac { 3 } { 2 } } ( 4 - x ) \quad x \geqslant 0$$ The point \(P\) is the stationary point of \(C\).

1. Find, using calculus, the \(x\) coordinate of \(P\). The region \(R _ { 1 }\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
   The region \(R _ { 2 }\), also shown shaded in Figure 1, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = k\), where \(k\) is a constant. Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
2. find, using calculus, the exact value of \(k\).

1. Show that the volume of the solid formed is \(\frac { 1 } { 4 } \pi ( \pi + 2 )\).
2. Find a cartesian equation for the curve.

8 \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-12_771_839_262_651} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 8\). The chord \(A B\) of the circle intersects the positive \(y\)-axis at \(A\) and is parallel to the \(x\)-axis.

1. Find, by calculation, the coordinates of \(A\) and \(B\).
2. Find the volume of revolution when the shaded segment, bounded by the circle and the chord \(A B\), is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

12. \begin{figure}[h] \end{figure} Figure 5 shows part of the curve \(C\) with parametric equations $$x = 2 \cos \theta \quad y = \sin 2 \theta \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$ The region \(R\), shown shaded in figure 5, is bounded by the curve \(C\), the line \(x = \sqrt { 2 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid revolution.
a. Show that the volume of the solid of revolution formed is given by the integral. $$k \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \sin ^ { 3 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$ where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-22_164_1148_54_118}
b. Hence, find the exact value for this volume, giving your answer in the form \(p \pi \sqrt { 2 }\) where \(p\) is a constant.

6. \includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-2_444_825_1571_516} The diagram shows the curve with equation \(y = \sqrt { \frac { x } { x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Use Simpson's rule with six strips to estimate the area of the shaded region. The shaded region is rotated through four right angles about the \(x\)-axis.
2. Show that the volume of the solid formed is \(\pi ( 3 - \ln 4 )\).

8 The equation of a curve is \(y = \frac { 6 } { 5 - 2 x }\).

1. Calculate the gradient of the curve at the point where \(x = 1\).
2. A point with coordinates \(( x , y )\) moves along the curve in such a way that the rate of increase of \(y\) has a constant value of 0.02 units per second. Find the rate of increase of \(x\) when \(x = 1\).
3. The region between the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume obtained is \(\frac { 12 } { 5 } \pi\).

9 \includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-10_540_1113_260_516} The diagram shows part of the curve with equation \(y ^ { 2 } = x - 2\) and the lines \(x = 5\) and \(y = 1\). The shaded region enclosed by the curve and the lines is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained.

15 A typical tube of toothpaste measures 5.4 cm across the straight edge at the top and is 12 cm high. It contains 75 ml of toothpaste so it needs to have an internal volume of \(75 \mathrm {~cm} ^ { 3 }\). Comment on the accuracy of the formula \(V = \frac { 2 } { 3 } \pi r ^ { 2 } h\), as given in line 41 , for the volume in this case. \section*{END OF QUESTION PAPER}

10 \includegraphics[max width=\textwidth, alt={}, center]{96cc217a-ffb3-4764-946e-e32271784ad7-4_764_929_255_609} The diagram shows the line \(y = x + 1\) and the curve \(y = \sqrt { } ( x + 1 )\), meeting at \(( - 1,0 )\) and \(( 0,1 )\).

1. Find the area of the shaded region.
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.