4.08d Volumes of revolution: about x and y axes

8. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central vertical cross-section, \(O A B C D E O\), of the design for a solid glass ornament. Figure 2 shows the finite region, \(R\), which is bounded by the \(y\)-axis, the horizontal line \(C B\), the vertical line \(B A\), and the curve \(A O\). The ornament is formed by rotating the region \(R\) through \(360 ^ { \circ }\) about the \(y\)-axis.
The curve \(A O\) is modelled by the equation $$x = k y ^ { 2 } + \sqrt { y } \quad 0 \leqslant y \leqslant 4$$ where \(k\) is a constant.
The point \(A\) has coordinates ( \(0.4,4\) ) and the point \(B\) has coordinates ( \(0.4,4.5\) )
The units are centimetres.

1. Determine the value of \(k\) according to this model.
2. Use algebraic integration to determine the exact volume of glass that would be required to make the ornament, according to the model.
3. State a limitation of the model. When the ornament was manufactured, \(9 \mathrm {~cm} ^ { 3 }\) of glass was required.
4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.

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.

7. \begin{figure}[h] \end{figure} Figure 2 shows the image of a gold pendant which has height 2 cm . The pendant is modelled by a solid of revolution of a curve \(C\) about the \(y\)-axis. The curve \(C\) has parametric equations $$x = \cos \theta + \frac { 1 } { 2 } \sin 2 \theta , \quad y = - ( 1 + \sin \theta ) \quad 0 \leqslant \theta \leqslant 2 \pi$$

1. Show that a Cartesian equation of the curve \(C\) is $$x ^ { 2 } = - \left( y ^ { 4 } + 2 y ^ { 3 } \right)$$
2. Hence, using the model, find, in \(\mathrm { cm } ^ { 3 }\), the volume of the pendant.

8. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central vertical cross section \(A B C D\) of a paddling pool that has a circular horizontal cross section. Measurements of the diameters of the top and bottom of the paddling pool have been taken in order to estimate the volume of water that the paddling pool can contain. Using these measurements, the curve \(B D\) is modelled by the equation $$y = \ln ( 3.6 x - k ) \quad 1 \leqslant x \leqslant 1.18$$ as shown in Figure 2.

1. Find the value of \(k\).
2. Find the depth of the paddling pool according to this model. The pool is being filled with water from a tap.
3. Find, in terms of \(h\), the volume of water in the pool when the pool is filled to a depth of \(h \mathrm {~m}\). Given that the pool is being filled at a constant rate of 15 litres every minute,
4. find, in \(\mathrm { cmh } ^ { - 1 }\), the rate at which the water level is rising in the pool when the depth of the water is 0.2 m .

7. \begin{figure}[h] \end{figure} A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student's design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = 1\), the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) through \(360 ^ { \circ }\) about the \(y\)-axis. The point \(A\) has coordinates ( \(1,0.5\) ) and the point \(B\) has coordinates ( \(0.5,2.5\) ) where the units are centimetres. The curve \(C _ { 1 }\) is modelled by the equation $$x = \frac { a } { y + b } \quad 0.5 \leqslant y \leqslant 2.5$$

1. Determine the value of \(a\) and the value of \(b\) according to the model. The curve \(C _ { 2 }\) is modelled to be an arc of the circle with centre \(( 0,3 )\).
2. Use calculus to determine the volume of plastic required to make the chess piece according to the model.

1. Solutions based entirely on graphical or numerical methods are not acceptable.

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \operatorname { arsinh } x \quad x \geqslant 0$$ and the straight line with equation \(y = \beta\) The line and the curve intersect at the point with coordinates \(( \alpha , \beta )\) Given that \(\beta = \frac { 1 } { 2 } \ln 3\)

1. show that \(\alpha = \frac { 1 } { \sqrt { 3 } }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve with equation \(y = \operatorname { arsinh } x\), the \(y\)-axis and the line with equation \(y = \beta\) The region \(R\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
2. Use calculus to find the exact value of the volume of the solid generated.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} John picked 100 berries from a plant.
The largest berry picked was approximately 2.8 cm long.
The shape of this berry is modelled by rotating the curve with equation $$16 x ^ { 2 } + 3 y ^ { 2 } - y \cos \left( \frac { 5 } { 2 } y \right) = 6 \quad x \geqslant 0$$ shown in Figure 2, about the \(y\)-axis through \(2 \pi\) radians, where the units are cm .
Given that the \(y\) intercepts of the curve are - 1.545 and 1.257 to four significant figures,

1. use algebraic integration to determine, according to the model, the volume of this berry. Given that the 100 berries John picked were then squeezed for juice,
2. use your answer to part (a) to decide whether, in reality, there is likely to be enough juice to fill a \(200 \mathrm {~cm} ^ { 3 }\) cup, giving a reason for your answer.

10. \begin{figure}[h] \end{figure} A solid playing piece for a board game is modelled by rotating the curve \(C\), shown in Figure 2, through \(2 \pi\) radians about the \(x\)-axis. The curve \(C\) has equation $$y = \sqrt { 1 + \frac { x ^ { 2 } } { 9 } } \quad - 4 \leqslant x \leqslant 4$$ with units as centimetres.

1. Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the playing piece is given by $$S = p \pi \int _ { - 4 } ^ { 4 } \sqrt { 81 + 10 x ^ { 2 } } \mathrm {~d} x + q \pi$$ where \(p\) and \(q\) are constants to be determined. Using the substitution \(x = \frac { 9 } { \sqrt { 10 } } \sinh u\), or another algebraic integration method, and showing all your working,
2. determine the total surface area of the playing piece, giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\)

8. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a French horn with a detachable bell section.
The shape of the bell section can be modelled by rotating an exponential curve through \(360 ^ { \circ }\) about the \(x\)-axis, where units are centimetres. The model uses the curve shown in Figure 2, with equation $$y = \frac { 9 } { 2 } e ^ { \frac { 1 } { 9 } x } \quad 0 \leqslant x \leqslant 9$$

1. Show that, according to this model, the external surface area of the bell section is given by $$K \int _ { 0 } ^ { 9 } \mathrm { e } ^ { \frac { 1 } { 9 } x } \sqrt { 4 + \mathrm { e } ^ { \frac { 2 } { 9 } x } } \mathrm {~d} x$$ where \(K\) is a real constant to be determined.
2. Use the substitution \(u = e ^ { \frac { 1 } { 9 } x }\) to show that $$\int _ { 0 } ^ { 9 } \mathrm { e } ^ { \frac { 1 } { 9 } x } \sqrt { 4 + \mathrm { e } ^ { \frac { 2 } { 9 } x } } \mathrm {~d} x = 9 \int _ { a } ^ { b } \frac { 2 u + u ^ { 3 } } { \sqrt { 4 u ^ { 2 } + u ^ { 4 } } } \mathrm {~d} u + 18 \int _ { a } ^ { b } \frac { 1 } { \sqrt { 4 + u ^ { 2 } } } \mathrm {~d} u$$ where \(a\) and \(b\) are constants to be determined. Hence, using algebraic integration,
3. determine, according to the model, the external surface area of the bell section of the horn, giving your answer to 3 significant figures.

6. \begin{figure}[h] \end{figure} The shaded region shown in Figure 4 is bounded by the \(x\)-axis, the line with equation \(x = 9\) and the line with equation \(y = \frac { 1 } { 3 } x\). This shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm . The cone is non-uniform and the mass per unit volume of the cone at the point ( \(x , y , z\) ) is \(\lambda x \mathrm {~kg} \mathrm {~cm} ^ { - 3 }\), where \(0 \leqslant x \leqslant 9\) and \(\lambda\) is constant.

1. Find the distance of the centre of mass of the cone from its vertex. A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm , so that the centres of the two plane surfaces coincide. The weight of the cone is \(W\) newtons and the weight of the hemisphere is \(k W\) newtons.
   When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
2. Find the value of \(k\)

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The shaded region shown in Figure 5 is bounded by the line with equation \(x = a\) and the curve with equation \(x ^ { 2 } + y ^ { 2 } = 4 a ^ { 2 }\) This shaded region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid is used to model a dome with height \(a\) metres and base radius \(\sqrt { 3 } a\) metres.
The dome is modelled as being non-uniform with the mass per unit volume of the dome at the point \(( x , y , z )\) equal to \(\frac { \lambda } { x ^ { 2 } } \mathrm {~kg} \mathrm {~m} ^ { - 3 }\), where \(a \leqslant x \leqslant 2 a\) and \(\lambda\) is a constant.

1. Show that the distance of the centre of mass of the dome from the centre of its plane face is \(\left( 4 \ln 2 - \frac { 5 } { 2 } \right) a\) metres. A solid uniform right circular cone has base radius \(\sqrt { 3 } a\) metres and perpendicular height \(4 a\) metres. A toy is formed by attaching the plane surface of the dome to the plane surface of the cone, as shown in Figure 6. The weight of the cone is \(k W\) and the weight of the dome is \(2 W\) The centre of mass of the toy is a distance \(d\) metres from the plane face of the dome.
2. Show that \(d = \frac { | k + 5 - 8 \ln 2 | } { 2 + k } a\) The toy is suspended from a point on the circumference of the plane face of the dome and hangs freely in equilibrium with the plane face of the dome at an angle \(\alpha\) to the downward vertical.
   Given that \(\tan \alpha = \frac { 1 } { 2 \sqrt { 3 } }\)
3. find the exact value of \(k\).

6. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The shaded region, shown in Figure 4, is bounded by the \(x\)-axis, the line with equation \(x = 6\), the line with equation \(y = 2\) and the \(y\)-axis. This region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { x }\)-axis to form a solid of revolution. This solid is used to model a non-uniform cylinder of height 6 cm and radius 2 cm . The mass per unit volume of the cylinder at the point \(( x , y , z )\) is \(\lambda ( x + 2 ) \mathrm { kg } \mathrm { cm } ^ { - 3 }\), where \(0 \leqslant x \leqslant 6\) and \(\lambda\) is a constant.

1. Show that the mass of the cylinder is \(120 \lambda \pi \mathrm {~kg}\).
2. Show that the centre of mass of the cylinder is 3.6 cm from \(O\). The point \(O\) is the centre of one end of the cylinder. The point \(A\) is the centre of the other end of the cylinder. A uniform solid hemisphere of radius 3 cm has density \(\lambda \mathrm { kg } \mathrm { cm } ^ { - 3 }\). The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point \(A\) on the cylinder to form the model shown in Figure 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_309_673_1713_696} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The model is placed with the end containing \(O\) on a rough inclined plane which is inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.
3. Find the value of \(\alpha\).

1 In this question you must show detailed reasoning.

1. Show that \(\cosh ( 2 \ln 3 ) = \frac { 41 } { 9 }\). The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \sqrt { \operatorname { sinhx } }\), the \(x\)-axis and the line with equation \(x = 2 \ln 3\) (see diagram). The units of the axes are centimetres. \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-2_652_668_740_242} A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the shape of the solid formed by rotating \(R\) completely about the \(x\)-axis.
2. Determine, according to the model, the exact volume of one chocolate piece.

6 In this question you must show detailed reasoning. As shown in Fig. 6.1, the region R is bounded by the lines \(x = 1 , x = 2 , y = 0\) and the curve \(y = 2 x ^ { 2 }\) for \(1 \leq x \leq 2\). A uniform solid of revolution, S , is formed when R is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h] \end{figure}

1. Show that the volume of S is \(\frac { 124 \pi } { 5 }\).
2. Show that the distance of the centre of mass of S from the centre of its smaller circular plane surface is \(\frac { 43 } { 62 }\). Fig. 6.2 shows S placed so that its smaller circular plane surface is in contact with a slope inclined at \(\alpha ^ { \circ }\) to the horizontal. S does not slip but is on the point of tipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_458_565_2014_694} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Find the value of \(\alpha\), giving your answer in degrees correct to 3 significant figures.

4 In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h] \end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.

4 In this question you must show detailed reasoning. The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }\), the \(x\)-axis and the lines with equations \(x = \frac { 1 } { 2 }\) and \(x = 1\) (see diagram). The units of the axes are cm . \includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when \(R\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.

10

1. (a) A curve has polar equation \(r = 2 - \sec \theta\). Show that the cartesian equation of the curve can be written in the form $$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$ The figure shows a sketch of part of the curve with equation \(y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}
   (b) Explain why the curve is symmetrical in the \(x\)-axis.
   (c) The line \(x = a\) is an asymptote of the curve. State the value of \(a\).
2. The enclosed loop shown in the figure is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. Find the exact volume of the solid formed. \section*{END OF QUESTION PAPER}

6 In this question you must show detailed reasoning. It is given that \(I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t\) for integers \(n \geqslant 0\).

1. Show that \(I _ { 1 } = \frac { 7 } { 3 }\).
2. Prove that, for \(n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }\). The curve \(C\) is defined parametrically by $$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$ When the curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
3. Determine

2 In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac { 8 } { \sqrt { 16 + x ^ { 2 } } }\), the \(x\)-axis and the lines \(x = 0\) and \(x = 4\). Region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume generated. [4]