4.08d Volumes of revolution: about x and y axes

\includegraphics{figure_1} Figure 1 shows part of a curve \(C\) with equation \(y = x^2 + 3\). The shaded region is bounded by \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 3\). The shaded region is rotated \(360°\) about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer as an exact multiple of \(\pi\). [7]

1. Use integration by parts to show that $$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2}\ln 2.$$ [6]

\includegraphics{figure_1} The finite region \(R\), bounded by the equation \(y = x^{\frac{1}{2}} \sec x\), the line \(x = \frac{\pi}{4}\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Find the volume of the solid of revolution generated. [2]
2. Find the gradient of the curve with equation \(y = x^{\frac{1}{2}} \sec x\) at the point where \(x = \frac{\pi}{4}\). [3]

\includegraphics{figure_1} In Fig. 1, the curve \(C\) has equation \(y = f(x)\), where $$f(x) = x + \frac{2}{x^2}, \quad x > 0.$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi(a + \ln b)\), where \(a\) and \(b\) are constants. [8]

\includegraphics{figure_2} 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 \(B\). [1]
2. Show that an equation of \(C\) is \(\frac{3y + 4}{y} = 0\), \(y \neq 0\). [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 \(360°\) about the \(y\)-axis to form a solid shape \(S\).

1. Find the volume of \(S\), giving your answer in the form \(\pi (a + b \ln c)\), where \(a\), \(b\) and \(c\) are integers. [7]

The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,

1. show that the volume of the tower is approximately \(15\,500\) m\(^3\). [2]

\includegraphics{figure_2} Figure 2 shows part of the curve with equation \(y = x^2 + 2\). The finite region \(R\) is bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).

1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [5]
2. State, with a reason, whether your answer in part \((a)\) is an under-estimate or over-estimate of the area of \(R\). [1]
3. Using integration, find the volume of the solid generated when \(R\) is rotated through \(360°\) about the \(x\)-axis, giving your answer in terms of \(\pi\). [6]

\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = 1 + \frac{1}{2\sqrt{x}}\). The shaded region \(R\), bounded by the curve, that \(x\)-axis and the lines \(x = 1\) and \(x = 4\), is rotated through \(360°\) about the \(x\)-axis. Using integration, show that the volume of the solid generated is \(\pi (5 + \frac{1}{2} \ln 2)\). [8]

A curve has equation $$y = \sqrt{9 - x^2} \quad 0 \leq x \leq 3$$

1. Using calculus, show that the length of the curve is \(\frac{3\pi}{2}\) [4]

The curve is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Using calculus, find the exact area of the surface generated. [3]

The curve \(C\) has parametric equations $$x = 3t^4, \quad y = 4t^3, \quad 0 \leq t \leq 1$$ The curve \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis. The area of the curved surface generated is \(S\).

1. Show that $$S = k\pi \int_{0}^{1} t^2(t^2 + 1)^{\frac{1}{2}} dt$$ where \(k\) is a constant to be found. [4]
2. Use the substitution \(u^2 = t^2 + 1\) to find the value of \(S\), giving your answer in the form \(p\pi\left(11\sqrt{2} - 4\right)\) where \(p\) is a rational number to be found. [7]

The curve \(C\) has equation \(y = 2x^3\), \(0 \leq x \leq 2\). The curve \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, find the area of the surface generated, giving your answer to 3 significant figures. [5]

A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x^2 + y^2 = r^2\) where \(r\) is a constant.

1. Show that \(1 + \left(\frac{dy}{dx}\right)^2 = \frac{r^2}{r^2 - x^2}\) [3]
2. Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4\pi r^2\). [5]
3. Write down the length of the arc of the curve \(y = \sqrt{1 - x^2}\) from \(x = 0\) to \(x = 1\) [1]

\includegraphics{figure_6} The curve \(C\) shown in Fig. 1 has equation \(y^2 = 4x\), \(0 \leq x \leq 1\). The part of the curve in the first quadrant is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Show that the surface area of the solid generated is given by $$4\pi \int_0^1 \sqrt{1+x} \, dx.$$ [4]
2. Find the exact value of this surface area. [3]
3. Show also that the length of the curve \(C\), between the points \((1, -2)\) and \((1, 2)\), is given by $$2 \int_0^1 \sqrt{\frac{x+1}{x}} \, dx.$$ [3]
4. Use the substitution \(x = \sinh^2 \theta\) to show that the exact value of this length is $$2[\sqrt{2} + \ln(1 + \sqrt{2})].$$ [6]

\includegraphics{figure_25} Figure 1 shows the curve with parametric equations $$x = a \cos^3 \theta, \quad y = a \sin^3 \theta, \quad 0 \leq \theta < 2\pi.$$

1. Find the total length of this curve. [7]

The curve is rotated through \(\pi\) radians about the \(x\)-axis.

1. Find the area of the surface generated. [5]

\includegraphics{figure_31} Figure 1 shows a sketch of the curve with parametric equations $$x = a \cos^3 t, \quad y = a \sin^3 t, \quad 0 \leq t \leq \frac{\pi}{2},$$ where \(a\) is a positive constant. The curve is rotated through \(2\pi\) radians about the \(x\)-axis. Find the exact value of the area of the curved surface generated. [8]

\includegraphics{figure_2} Figure 2 shows the region \(R\) bounded by the curve with equation \(y^2 = rx\), where \(r\) is a positive constant, the \(x\)-axis and the line \(x = r\). A uniform solid of revolution \(S\) is formed by rotating \(R\) through one complete revolution about the \(x\)-axis.

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac{4}{5}r\). [6]

The solid is placed with its plane face on a plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. Given that \(S\) does not topple,

1. find, to the nearest degree, the maximum value of \(\alpha\). [4]

\includegraphics{figure_2} The region \(R\) is bounded by the curve with equation \(y = e^x\), the line \(x = 1\), the line \(x = 2\) and the \(x\)-axis as shown in Figure 2. A uniform solid \(S\) is formed by rotating \(R\) through \(2\pi\) about the \(x\)-axis.

1. Show that the volume of \(S\) is \(\frac{1}{2}\pi (e^4 - e^2)\). [4]
2. Find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of \(S\). [6]

\includegraphics{figure_3} The shaded region \(R\) is bounded by part of the curve with equation \(y = \frac{1}{4}(x - 2)^2\), the \(x\)-axis and the \(y\)-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm. A uniform solid \(S\) is made by rotating \(R\) through \(360°\) about the \(x\)-axis. Using integration,

1. calculate the volume of the solid \(S\), leaving your answer in terms of \(\pi\), [4]
2. show that the centre of mass of \(S\) is \(\frac{4}{5}\) cm from its plane face. [7]

\includegraphics{figure_4} A tool is modelled as having two components, a solid uniform cylinder \(C\) and the solid \(S\). The diameter of \(C\) is 4 cm and the length of \(C\) is 8 cm. One end of \(C\) coincides with the plane face of \(S\). The components are made of different materials. The weight of \(C\) is \(10W\) newtons and the weight of \(S\) is \(2W\) newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports \(A\) and \(B\), which are at the ends of the cylinder, as shown in Fig. 4.

1. Find the magnitude of the force of the support \(A\) on the tool. [5]

A uniform solid is formed by rotating the region enclosed between the curve with equation \(y = \sqrt{x}\), the \(x\)-axis and the line \(x = 4\), through one complete revolution about the \(x\)-axis. Find the distance of the centre of mass of the solid from the origin \(O\). [5]

The finite region bounded by the \(x\)-axis, the curve \(y = \frac{1}{x}\), the line \(x = \frac{1}{4}\) and the line \(x = 1\), is rotated through one complete revolution about the \(x\)-axis to form a uniform solid of revolution.

1. Show that the volume of the solid is \(21\pi\). [4]
2. Find the coordinates of the centre of mass of the solid. [5]

\includegraphics{figure_9} The diagram shows the curve with equation \(y = 2 \ln(x - 1)\). The point \(P\) has coordinates \((0, p)\). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = p\). The units on the axes are centimetres. The region \(R\) is rotated completely about the \(y\)-axis to form a solid.

1. Show that the volume, \(V \text{ cm}^3\), of the solid is given by $$V = \pi(e^p + 4e^{\frac{p}{2}} + p - 5).$$ [8]
2. It is given that the point \(P\) is moving in the positive direction along the \(y\)-axis at a constant rate of \(0.2 \text{ cm min}^{-1}\). Find the rate at which the volume of the solid is increasing at the instant when \(p = 4\), giving your answer correct to 2 significant figures. [5]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{1}{\sqrt{3x + 2}}\). The shaded region is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\).

1. Find the exact area of the shaded region. [4]
2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed, simplifying your answer. [5]

\includegraphics{figure_5} The diagram shows the curve \(y = \frac{6}{\sqrt{3x + 1}}\). The shaded region is bounded by the curve and the lines \(x = 2\), \(x = 9\) and \(y = 0\).

1. Show that the area of the shaded region is \(4\sqrt{7}\) square units. [4]
2. The shaded region is rotated completely about the \(x\)-axis. Show that the volume of the solid produced can be written in the form \(k\ln 2\), where the exact value of the constant \(k\) is to be determined. [5]

\includegraphics{figure_2} The diagram shows the curve with equation \(y = (2x - 3)^2\). The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\). Find the exact volume obtained when the shaded region is rotated completely about the \(x\)-axis. [5]

\includegraphics{figure_4} The diagram shows part of the curve \(y = \frac{k}{x}\), where \(k\) is a positive constant. The points A and B on the curve have \(x\)-coordinates 2 and 6 respectively. Lines through A and B parallel to the axes as shown meet at the point C. The region R is bounded by the curve and the lines \(x = 2\), \(x = 6\) and \(y = 0\). The region S is bounded by the curve and the lines AC and BC. It is given that the area of the region R is \(\ln 81\).

1. Show that \(k = 4\). [3]
2. Find the exact volume of the solid produced when the region S is rotated completely about the \(x\)-axis. [4]