Volume with implicit or parametric curves

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. \begin{figure}[h] \end{figure} Figure 1 shows the curve with parametric equations $$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leq t < \pi .$$ The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).

1. Find the value of \(t\) at \(O\) and the value of \(t\) at \(A\). The region enclosed by the curve is rotated through \(\pi\) radians about the \(x\)-axis.
2. Show that the volume of the solid formed is given by $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 12 \pi \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$
3. Using the substitution \(u = \sin t\), or otherwise, evaluate this integral, giving your answer as an exact multiple of \(\pi\).
   6. continued

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

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A large pile of concrete waste is created on a building site.
Figure 1 shows a central vertical cross-section of the concrete waste.
The curve \(C\), shown in Figure 2, has equation $$y + x ^ { 2 } = 2 \quad 0 \leqslant x \leqslant \sqrt { 2 }$$ The region \(R\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the \(x\)-axis and the curve \(C\). The volume of concrete waste is modelled by the volume of revolution formed when \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. The units are metres. The density of the concrete waste is \(900 \mathrm { kgm } ^ { - 3 }\)

1. Use the model to estimate the mass of the concrete waste. Give your answer to 2 significant figures.
2. Give a limitation of the model. The mass of the concrete waste is approximately 5500 kg .
3. Use this information and your answer to part (a) to evaluate the model, giving a reason for your answer.

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.

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.

9. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central vertical cross-section \(A B C D E F A\) of a vase together with measurements that have been taken from the vase. The horizontal cross-section between \(A B\) and \(F C\) is a circle with diameter 4 cm .
The base of the vase \(E D\) is horizontal and the point \(E\) is vertically below \(F\) and the point \(D\) is vertically below \(C\). Using these measurements, the curve \(C D\) is modelled by the parametric equations $$x = a + 3 \sin 2 t \quad y = b \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Determine the value of \(a\) and the value of \(b\) according to the model.
2. Using algebraic integration and showing all your working, determine, according to the model, the volume of the vase, giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\)
3. State a limitation of the model.

10 The diagram below shows an ellipse \(E\) The coordinate axes are the lines of symmetry of \(E\) \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-14_645_780_450_630} 10

1. Write down an equation of \(E\) 10
2. The region bounded by the \(x\)-axis and the ellipse \(E\) for \(y \geq 0\) is shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-15_643_775_408_635} A solid \(S\) is formed by rotating the shaded region through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of \(S\) is \(a \pi\) where \(a\) is an integer to be found. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-16_2488_1732_219_139}