4.08d Volumes of revolution: about x and y axes

4 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where \(x = 0\) to the point where \(x = 1\) is denoted by \(s\). Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { } \left( 1 + x ^ { 4 } \right) \mathrm { d } x$$ The surface area generated when this arc is rotated through one complete revolution about the \(x\)-axis is denoted by \(S\). Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { } 2 - 1 )$$ [Do not attempt to evaluate \(s\) or \(S\).]

3 At any point \(( x , y )\) on the curve \(C\), $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t \sqrt { } \left( t ^ { 2 } + 4 \right) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = - t \sqrt { } \left( 4 - t ^ { 2 } \right)$$ where the parameter \(t\) is such that \(0 \leqslant t \leqslant 2\). Show that the length of \(C\) is \(4 \sqrt { } 2\). Given that \(y = 0\) when \(t = 2\), determine the area of the surface generated when \(C\) is rotated through one complete revolution about the \(x\)-axis, leaving your answer in an exact form.

4 The parametric equations of a curve are $$x = \cos t + t \sin t , \quad y = \sin t - t \cos t$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \frac { 1 } { 2 } \pi\) is rotated about the \(x\)-axis through one complete revolution. Find the area of the surface generated, leaving your result in terms of \(\pi\).

6 The curve \(C\) has parametric equations $$x = \mathrm { e } ^ { t } - 4 t + 3 , \quad y = 8 \mathrm { e } ^ { \frac { 1 } { 2 } t } , \quad \text { for } 0 \leqslant t \leqslant 2$$

1. Find, in terms of e , the length of \(C\).
2. Find, in terms of \(\pi\) and e , the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

A curve \(C\) has parametric equations $$x = \mathrm { e } ^ { 2 t } \cos 2 t , \quad y = \mathrm { e } ^ { 2 t } \sin 2 t , \quad \text { for } - \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi .$$ Find the arc length of \(C\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

The curve \(C\) has equation \(y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) for \(0 \leqslant x \leqslant 4\).

1. The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = 4\). Find, in terms of e, the coordinates of the centroid of the region \(R\).
2. Show that \(\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\), where \(s\) denotes the arc length of \(C\), and find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

5 A curve \(C\) has parametric equations $$x = \frac { 2 } { 5 } t ^ { \frac { 5 } { 2 } } - 2 t ^ { \frac { 1 } { 2 } } , \quad y = \frac { 4 } { 3 } t ^ { \frac { 3 } { 2 } } , \quad \text { for } 1 \leqslant t \leqslant 4$$

1. Find the exact value of the arc length of \(C\).
2. Find also the exact value of the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

5 A curve \(C\) is defined parametrically by $$x = \frac { 2 } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } } \quad \text { and } \quad y = \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } }$$ for \(0 \leqslant t \leqslant 1\). The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).

1. Show that \(S = 4 \pi \int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \left( \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } \right) ^ { 2 } } \mathrm {~d} t\).
2. Using the substitution \(u = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }\), or otherwise, find \(S\) in terms of \(\pi\) and e .

2 The curve \(C\) is defined parametrically by $$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t , \quad 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ where \(a\) is a positive constant. Find the area of the surface generated when \(C\) is rotated through one complete revolution about the \(x\)-axis.

The curve \(C\) has equation $$y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } } + \lambda ,$$ where \(\lambda > 0\) and \(0 \leqslant x \leqslant 3\). The length of \(C\) is denoted by \(s\). Prove that \(s = 2 \sqrt { } 3\). The area of the surface generated when \(C\) is rotated through one revolution about the \(x\)-axis is denoted by \(S\). Find \(S\) in terms of \(\lambda\). The \(y\)-coordinate of the centroid of the region bounded by \(C\), the axes and the line \(x = 3\) is denoted by h. Given that \(\int _ { 0 } ^ { 3 } y ^ { 2 } \mathrm {~d} x = \frac { 3 } { 4 } + \frac { 8 \sqrt { } 3 } { 5 } \lambda + 3 \lambda ^ { 2 }\), show that $$\lim _ { \lambda \rightarrow \infty } \frac { S } { h s } = 4 \pi$$

The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { \frac { 1 } { 2 } } ( 3 - x )\), for \(0 \leqslant x \leqslant 3\). Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 3\). Show that $$\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( x ^ { - \frac { 1 } { 2 } } + x ^ { \frac { 1 } { 2 } } \right)$$ where \(s\) denotes arc length, and find the arc length of \(C\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

8 The curve \(C\) has parametric equations $$x = \frac { 1 } { 3 } t ^ { 3 } - \ln t , \quad y = \frac { 4 } { 3 } t ^ { \frac { 3 } { 2 } }$$ for \(1 \leqslant t \leqslant 3\). Find the arc length of \(C\). Find also the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

1 The curve \(C\) has equation \(y = x ^ { a }\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis.

3

1. 1. Given that \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \left( \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } + \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) ^ { 2 }\). The arc of the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln a\) (where \(a > 1\) ) is denoted by \(C\).
   2. Show that the length of \(C\) is \(\frac { a - 1 } { \sqrt { a } }\).
   3. Find the area of the surface formed when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. An ellipse has parametric equations \(x = 2 \cos \theta , y = \sin \theta\) for \(0 \leqslant \theta < 2 \pi\).
   1. Show that the normal to the ellipse at the point with parameter \(\theta\) has equation $$y = 2 x \tan \theta - 3 \sin \theta$$
   2. Find parametric equations for the evolute of the ellipse, and show that the evolute has cartesian equation $$( 2 x ) ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 3 ^ { \frac { 2 } { 3 } }$$
   3. Using the evolute found in part (ii), or otherwise, find the radius of curvature of the ellipse
      (A) at the point \(( 2,0 )\),
      (B) at the point \(( 0,1 )\).

1 A curve is defined parametrically by $$x = a t ^ { 2 } , \quad y = a t$$ where \(a\) is a positive constant. The part of the curve joining the point where \(t = 0\) to the point where \(t = \sqrt { } 2\) is rotated through one complete revolution about the \(x\)-axis. Show that the area of the surface obtained is \(\frac { 13 } { 3 } \pi a ^ { 2 }\).

6 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { 4 } t ^ { 4 } - \ln t$$ for \(1 \leqslant t \leqslant 2\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(y\)-axis.

4 Fig. 4 shows a sketch of the region enclosed by the curve \(\sqrt { 1 + \mathrm { e } ^ { - 2 x } }\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\). \begin{figure}[h] \end{figure} Find the volume of the solid generated when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in an exact form.

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.

8

1. Given that \(\mathrm { e } ^ { - 2 x } = 3\), find the exact value of \(x\).
2. Use integration by parts to find \(\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
3. A curve has equation \(y = \mathrm { e } ^ { - 2 x } + 6 x\).
   1. Find the exact values of the coordinates of the stationary point of the curve.
   2. Determine the nature of the stationary point.
   3. The region \(R\) is bounded by the curve \(y = \mathrm { e } ^ { - 2 x } + 6 x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.

8

1. Given that \(\mathrm { e } ^ { - 2 x } = 4\), find the exact value of \(x\).
2. The diagram shows the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\). \includegraphics[max width=\textwidth, alt={}, center]{6761e676-48ae-47e9-9617-153342cdf5c4-9_490_1185_463_440} The curve crosses the \(y\)-axis at the point \(A\), the \(x\)-axis at the point \(B\), and has a stationary point at \(M\).
   1. State the \(y\)-coordinate of \(A\).
   2. Find the \(x\)-coordinate of \(B\), giving your answer in an exact form.
   3. Find the \(x\)-coordinate of the stationary point, \(M\), giving your answer in an exact form.
   4. The shaded region \(R\) is bounded by the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\), the lines \(x = 0\) and \(x = \ln 2\) and the \(x\)-axis. Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in the form \(\frac { p } { q } \pi\), where \(p\) and \(q\) are integers.
      (7 marks)

7

1. A curve has equation \(y = x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } }\).
   Show that the curve has exactly two stationary points and find the exact values of their coordinates.
   (7 marks)
2. 1. Use integration by parts twice to find the exact value of \(\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } } \mathrm {~d} x\).
   2. The region bounded by the curve \(y = 3 x \mathrm { e } ^ { - \frac { x } { 8 } }\), the \(x\)-axis from 0 to 4 and the line \(x = 4\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid. Use your answer to part (b)(i) to find the exact value of the volume of the solid generated.

6

1. 1. Sketch the graph of \(y = 4 - x ^ { 2 }\), indicating the coordinates of the points where the graph crosses the coordinate axes.
   2. The region between the graph and the \(x\)-axis from \(x = 0\) to \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume of the solid generated.
2. 1. Sketch the graph of \(y = \left| 4 - x ^ { 2 } \right|\).
   2. Solve \(\left| 4 - x ^ { 2 } \right| = 3\).
   3. Hence, or otherwise, solve the inequality \(\left| 4 - x ^ { 2 } \right| < 3\).

7

1. Given that \(z = \frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } z } { \mathrm {~d} x } = \sec ^ { 2 } x\).
2. Sketch the curve with equation \(y = \sec x\) for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
3. The region \(R\) is bounded by the curve \(y = \sec x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.

6 The diagram shows the curve with equation \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) for \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-5_483_611_402_717}

1. Find the gradient of the curve \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) at the point where \(x = \ln 2\).
2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 2 } \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
3. The shaded region \(R\) is bounded by the curve, the lines \(x = 0 , x = 2\) and the \(x\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

6 The diagram shows the curve with equation \(y = \sqrt { 100 - 4 x ^ { 2 } }\), where \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_518_494_367_758}

1. Calculate the volume of the solid generated when the region bounded by the curve shown above and the coordinate axes is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis, giving your answer in terms of \(\pi\).
2. Use the mid-ordinate rule with five strips of equal width to find an estimate for \(\int _ { 0 } ^ { 5 } \sqrt { 100 - 4 x ^ { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
3. The point \(P\) on the curve has coordinates \(( 3,8 )\).
   1. Find the gradient of the curve \(y = \sqrt { 100 - 4 x ^ { 2 } }\) at the point \(P\).
   2. Hence show that the equation of the tangent to the curve at the point \(P\) can be written as \(2 y + 3 x = 25\).
4. The shaded regions on the diagram below are bounded by the curve, the tangent at \(P\) and the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_642_546_1800_731} Use your answers to part (b) and part (c)(ii) to find an approximate value for the total area of the shaded regions. Give your answer to three significant figures.