Volume of revolution (parametric)

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan t , \quad y = 2 \sin ^ { 2 } t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The finite region \(S\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(x = \sqrt { 3 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.

1. Show that the volume of the solid of revolution formed is given by $$4 \pi \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \tan ^ { 2 } t - \sin ^ { 2 } t \right) \mathrm { d } t$$
2. Hence use integration to find the exact value for this volume.

10. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { 3 } , \frac { 1 } { 2 } \sqrt { 3 } \right)\)

1. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
2. Show that \(Q\) has coordinates \(( k \sqrt { 3 } , 0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { 3 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { 3 } + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-39_61_29_2608_1886}

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = \tan \theta \quad y = 2 \sin 2 \theta \quad \theta \geqslant 0$$ The finite region, shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \sqrt { 3 }\) The region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.

1. Show that the exact volume of this solid of revolution is given by $$\int _ { 0 } ^ { k } p ( 1 - \cos 2 \theta ) d \theta$$ where \(p\) and \(k\) are constants to be found.
2. Hence find, by algebraic integration, the exact volume of this solid of revolution.

1. The curve \(C\) has parametric equations

$$x = \ln t , \quad y = t ^ { 2 } - 2 , \quad t > 0$$ Find

1. an equation of the normal to \(C\) at the point where \(t = 3\),
2. a cartesian equation of \(C\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a3ece8a8-8107-4c3a-a6a9-c19b5e35ec5a-10_579_759_740_571} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} The finite area \(R\), shown in Figure 1, is bounded by \(C\), the \(x\)-axis, the line \(x = \ln 2\) and the line \(x = \ln 4\). The area \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
3. Use calculus to find the exact volume of the solid generated.

8. (a) Using the identity \(\cos 2 \theta = 1 - 2 \sin ^ { 2 } \theta\), find \(\int \sin ^ { 2 } \theta \mathrm {~d} \theta\). \begin{figure}[h] \end{figure} Figure 4 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = 2 \sin 2 \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The finite shaded region \(S\) shown in Figure 4 is bounded by \(C\), the line \(x = \frac { 1 } { \sqrt { 3 } }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
(b) Show that the volume of the solid of revolution formed is given by the integral $$k \int _ { 0 } ^ { \frac { \pi } { 6 } } \sin ^ { 2 } \theta \mathrm {~d} \theta$$ where \(k\) is a constant.
(c) Hence find the exact value for this volume, giving your answer in the form \(p \pi ^ { 2 } + q \pi \sqrt { } 3\), where \(p\) and \(q\) are constants.

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { } 3 , \frac { 1 } { 2 } \sqrt { } 3 \right)\).

1. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
2. Show that \(Q\) has coordinates \(( k \sqrt { } 3,0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { } 3\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { } 3 + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants.

7. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3 \tan \theta , \quad y = 4 \cos ^ { 2 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates ( 3,2 ). The line \(l\) is the normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).

1. Find the \(x\) coordinate of the point \(Q\). The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This shaded region is rotated \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Find the exact value of the volume of the solid of revolution, giving your answer in the form \(p \pi + q \pi ^ { 2 }\), where \(p\) and \(q\) are rational numbers to be determined.
   [0pt] [You may use the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = t + \frac { 1 } { t } \quad y = t - \frac { 1 } { t } \quad t > 0.7$$ The curve \(C\) intersects the \(x\)-axis at the point \(Q\).

1. Find the \(x\) coordinate of \(Q\). The line \(l\) is the normal to \(C\) at the point \(P\) as shown in Figure 2.
   Given that \(t = 2\) at \(P\)
2. write down the coordinates of \(P\)
3. Using calculus, show that an equation of \(l\) is $$3 x + 5 y = 15$$ The region, \(R\), shown shaded in Figure 2 is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis.
4. Using algebraic integration, find the exact volume of the solid of revolution formed when the region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with parametric equations $$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The finite region \(S\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { 3 } { 2 }\) The shaded region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.

1. Show that the volume of the solid of revolution is given by $$k \int _ { 0 } ^ { a } \sin ^ { 2 } t \cos ^ { 3 } t \mathrm {~d} t$$ where \(k\) and \(a\) are constants to be given in terms of \(\pi\).
2. Use the substitution \(u = \sin t\), or otherwise, to find the exact value of this volume, giving your answer in the form \(\frac { p \pi } { q }\) where \(p\) and \(q\) are integers. (Solutions based entirely on graphical or numerical methods are not acceptable.)

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve with parametric equations $$x = t ^ { 3 } + 1 , \quad y = \frac { 2 } { t } , \quad t > 0 .$$ The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 9\).

1. Find the area of the shaded region.
2. Show that the volume of the solid formed when the shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis is \(12 \pi\).
3. Find a cartesian equation for the curve in the form \(y = \mathrm { f } ( x )\).
   7. continued

17. Figure 2
\includegraphics[max width=\textwidth, alt={}, center]{615ec68b-3a32-4309-bb54-acf39ed09f96-12_674_776_330_541} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac { 4 } { x - 3 } , x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\).
2. Show that an equation of \(C\) is \(\frac { 3 y + 4 } { y } , y \neq 0\).
   (1) The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis. The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis to form a solid shape \(S\).
3. Find the volume of \(S\), giving your answer in the form \(\pi ( a + b \ln c )\), where \(a , b\) and \(c\) are integers. The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
4. show that the volume of the tower is approximately \(15500 \mathrm {~m} ^ { 3 }\).