4.08d Volumes of revolution: about x and y axes

4 The curve \(C\) has parametric equations \(x = \frac { 1 } { 2 } t ^ { 2 } - \ln t , y = 2 t\), for \(1 \leqslant t \leqslant 4\). When \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed of surface area \(S\). Determine the exact value of \(S\).

\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ 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. [(a)] Find the \(x\) coordinate of the point \(Q\). \hfill [6]

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.

1. [(b)] 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. [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] \hfill [9] \end{enumerate} \end{enumerate}

\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \frac{1}{(5x - 4)^3}\) and the lines \(x = 2.4\) and \(y = 1\). The curve intersects the line \(y = 1\) at the point \((1, 1)\). Find the exact volume of the solid generated when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]

\includegraphics{figure_2} The diagram shows part of the curve \(y = \frac{a}{x}\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis is \(24\pi\), find the value of \(a\). [4]

\includegraphics{figure_11} The diagram shows part of the curve \(y = 4\sqrt{x} - x\). The curve has a maximum point at \(M\) and meets the \(x\)-axis at \(O\) and \(A\).

1. Find the coordinates of \(A\) and \(M\). [5]
2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis, giving your answer in terms of \(\pi\). [6]

\includegraphics{figure_1} The diagram shows the region enclosed by the curve \(y = \frac{6}{2x - 3}\), the \(x\)-axis and the lines \(x = 2\) and \(x = 3\). Find, in terms of \(\pi\), the volume obtained when this region is rotated through \(360°\) about the \(x\)-axis. [4]

\includegraphics{figure_5} The diagram shows part of the curve \(x = \frac{8}{y^2} - 2\), crossing the \(y\)-axis at the point \(A\). The point \(B (6, 1)\) lies on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\). Find the exact volume obtained when this shaded region is rotated through \(360°\) about the \(y\)-axis. [6]

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

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 \(360°\) about the \(x\)-axis. [4]
2. Given that the line \(2y = x + c\) is a normal to the curve, find the possible values of the constant \(c\). [6]

\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \sqrt{x^3 + x^2}\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [4]
2. \(P\) is the point on the curve with \(x\)-coordinate \(3\). Find the \(y\)-coordinate of the point where the normal to the curve at \(P\) crosses the \(y\)-axis. [6]

\includegraphics{figure_10} 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. [5]
2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [7]

\includegraphics{figure_1} The diagram shows part of the curve \(y = x^2 + 1\). Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [4]

A curve has equation \(y = (kx - 3)^{-1} + (kx - 3)\), where \(k\) is a non-zero constant.

1. Find the \(x\)-coordinates of the stationary points in terms of \(k\), and determine the nature of each stationary point. Justify your answers. [7]

1. \includegraphics{figure_3} The diagram shows part of the curve for the case when \(k = 1\). Showing all necessary working, find the volume obtained when the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), shown shaded in the diagram, is rotated through \(360°\) about the \(x\)-axis. [5]

\includegraphics{figure_4} The diagram shows the curve with equation \(y = \sqrt{1 + e^{0.5x}}\). The shaded region is bounded by the curve and the straight lines \(x = 0\), \(x = 6\) and \(y = 0\).

1. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures. [3]
2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [4]

\includegraphics{figure_5} The diagram shows the curve \(y = \sqrt{1 + e^{4x}}\) for \(0 \leq x \leq 6\). The region bounded by the curve and the lines \(x = 0\), \(x = 6\) and \(y = 0\) is denoted by \(R\).

1. Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 2 decimal places. [3]
2. With reference to the diagram, explain why this estimate is greater than the exact area of \(R\). [1]
3. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [4]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \sqrt{1 + 3\cos^2(\frac{1}{2}x)}\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).

1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures. [3]
2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced. [5]

\includegraphics{figure_11} The diagram shows part of the curve \(y = \frac{x}{2} + \frac{6}{x}\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).

1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\). [6]
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through 360° about the \(x\)-axis. Give your answer in terms of \(\pi\). [6]

\includegraphics{figure_11} The diagram shows part of the curve \(y = (x + 1)^2 + (x + 1)^{-1}\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2(x + 1)^3 = 1\) and find the exact value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) at \(A\). [5]
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]

Answer only one of the following two alternatives. EITHER The curve \(C\) has equation \(y = \frac{5(x-1)(x+2)}{(x-2)(x+3)}\).

1. Express \(y\) in the form \(P + \frac{Q}{x-2} + \frac{R}{x+3}\). [3]
2. Show that \(\frac{dy}{dx} = 0\) for exactly one value of \(x\) and find the corresponding value of \(y\). [4]
3. Write down the equations of all the asymptotes of \(C\). [3]
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\). [4]

OR A curve has equation \(y = \frac{5}{3}x^{\frac{3}{2}}\), for \(x \geq 0\). The arc of the curve joining the origin to the point where \(x = 3\) is denoted by \(R\).

1. Find the length of \(R\). [4]
2. Find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis, the line \(x = 3\) and \(R\). [5]
3. Show that the area of the surface generated when \(R\) is rotated through one revolution about the \(y\)-axis is \(\frac{232\pi}{15}\). [5]

A curve is defined parametrically by $$x = t - \frac{1}{2}\sin 2t \quad \text{and} \quad y = \sin^2 t.$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).

1. Show that $$S = a\pi \int_0^\pi \sin^3 t \, dt,$$ where the constant \(a\) is to be found. [5]
2. Using the result \(\sin 3t = 3\sin t - 4\sin^3 t\), find the exact value of \(S\). [3]

Answer only one of the following two alternatives. EITHER The curve \(C\) is defined parametrically by $$x = 18t - t^2 \quad \text{and} \quad y = 8t^{\frac{1}{2}},$$ where \(0 < t \leqslant 4\).

1. Show that at all points of \(C\), $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = \frac{-3(9 + t)}{2t^2(9 - t)^3}.$$ [4]
2. Show that the mean value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) with respect to \(x\) over the interval \(0 < x \leqslant 56\) is \(\frac{3}{70}\). [4]
3. Find the area of the surface generated when \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, showing full working. [6]

OR Let \(I_n = \int_1^{\sqrt{2}} (x^2 - 1)^n \mathrm{d}x\).

1. Show that, for \(n \geqslant 1\), $$(2n + 1)I_n = \sqrt{2} - 2nI_{n-1}.$$ [5]
2. Using the substitution \(x = \sec \theta\), show that $$I_n = \int_0^{\frac{1}{4}\pi} \tan^{2n+1} \theta \sec \theta \, \mathrm{d}\theta.$$ [4]
3. Deduce the exact value of $$\int_0^{\frac{1}{4}\pi} \frac{\sin^7 \theta}{\cos^8 \theta} \, \mathrm{d}\theta.$$ [5]

The curve \(C\) has parametric equations $$x = \frac{1}{2}t^2 - \ln t, \quad y = 2t + 1, \quad \text{for } \frac{1}{2} \leqslant t \leqslant 2.$$

1. Find the exact length of \(C\). [5]
2. Find \(\frac{d^2y}{dx^2}\) in terms of \(t\), simplifying your answer. [4]

\includegraphics{figure_3} The curve \(C\), shown in Figure 3, has equation $$y = \frac{x^{-\frac{1}{4}}}{\sqrt{1+x}\left(\arctan\sqrt{x}\right)}$$ 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 \(360°\) 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 $$k \int_a^b \frac{1}{u^2} du$$ where \(k\), \(a\) and \(b\) are constants to be found. [6]
2. Hence, using algebraic integration, find the value of \(V\) in simplest form. [3]

\includegraphics{figure_2} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan \theta, \quad y = 1 + 2\cos 2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The curve \(C\) crosses the \(x\)-axis at \((\sqrt{3}, 0)\). The finite shaded region \(S\) shown in Figure 2 is bounded by \(C\), the line \(x = 1\) 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 the integral $$k \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} (16 \cos^2 \theta - 8 + \sec^2 \theta) \, d\theta$$ where \(k\) is a constant. [5]
2. Hence, use integration to find the exact value for this volume. [5]

\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = 3^x$$ The point \(P\) lies on \(C\) and has coordinates \((2, 9)\). The line \(l\) is a tangent to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).

1. Find the exact value of the \(x\) coordinate of \(Q\). [4]

The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This region \(R\) is rotated through \(360°\) about the \(x\)-axis.

1. Use integration to find the exact value of the volume of the solid generated. Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are exact constants. [You may assume the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] [6]