4.08d Volumes of revolution: about x and y axes

1. Express \(17\cosh x - 15\sinh x\) in the form \(e^{-x}(ae^{bx} + c)\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]

A function is defined by \(f(x) = \frac{1}{\sqrt{17\cosh x - 15\sinh x}}\). The region bounded by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).

1. In this question you must show detailed reasoning. Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k\pi\tan^{-1} q\) where \(k\) and \(q\) are rational numbers to be determined. [7]

In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac{8}{\sqrt{16+x^3}}\), the \(x\)-axis and the lines \(x=0\) and \(x=4\). Region \(R\) is rotated through \(360°\) about the \(x\)-axis. Find the exact value of the volume generated. [4]

In this question you must show detailed reasoning.

1. Given that \(y = \arctan x\), show that \(\frac{dy}{dx} = \frac{1}{1+x^2}\). [3]

Fig. 12 shows the curve \(y = \frac{1}{1+x^2}\). \includegraphics{figure_12}

1. Find, in exact form, the mean value of the function \(f(x) = \frac{1}{1+x^2}\) for \(-1 \leq x \leq 1\). [3]
2. The region bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = -1\) is rotated through \(2\pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated. [7]

In this question you must show detailed reasoning. \includegraphics{figure_7} Fig. 7 shows the curve with equation \(y = \frac{2}{3}\ln x\). The region R, shown shaded in Fig. 7, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = \ln 2\). A uniform solid of revolution is formed by rotating the region R completely about the \(y\)-axis. Find the exact \(x\)-coordinate of the centre of mass of the solid. [8]

The region \(R\) is bounded by the curve \(x = \sin y\), the \(y\)-axis and the lines \(y = 1\), \(y = 3\). Find the volume of the solid generated when \(R\) is rotated through four right angles about the \(y\)-axis. Give your answer correct to two decimal places. [5]

The function \(f\) is defined by $$f(x) = \frac{1}{\sqrt{x^2 + 4x + 3}}.$$

1. Find the mean value of the function \(f\) for \(0 \leqslant x \leqslant 2\), giving your answer correct to three decimal places. [5]
2. The region \(R\) is bounded by the curve \(y = f(x)\), the \(x\)-axis and the lines \(x = 0\) and \(x = 2\). Find the exact value of the volume of the solid generated when \(R\) is rotated through four right angles about the \(x\)-axis. [6]

The curve \(y = 1 + x^3\) is denoted by \(C\).

1. A bowl is designed by rotating the arc of \(C\) joining the points \((0,1)\) and \((2,9)\) through four right angles about the \(y\)-axis. Calculate the capacity of the bowl. [5]
2. Another bowl with capacity 25 is to be designed by rotating the arc of \(C\) joining the points with \(y\) coordinates 1 and \(a\) through four right angles about the \(y\)-axis. Calculate the value of \(a\). [5]

\includegraphics{figure_4} The diagram shows the curves \(y = e^{3x}\) and \(y = (2x - 1)^4\). The shaded region is bounded by the two curves and the line \(x = \frac{1}{2}\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [7]

The region \(R\) between the \(x\)-axis, the curve \(y = \frac{1}{\sqrt{p + x^3}}\) and the lines \(x = \sqrt{p}\) and \(x = \sqrt{3p}\), where \(p\) is a positive parameter, is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).

1. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\). [5]
2. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt{48}\) find in exact form

\includegraphics{figure_1} Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = xe^x\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis. The region \(R\) is rotated through 360 degrees about the \(x\)-axis. Use integration by parts to find an exact value for the volume of the solid generated. [7]

Please remember to show detailed reasoning in your answer \includegraphics{figure_9} 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]

O is the origin of a coordinate system whose units are cm. The points A, B, C and D have coordinates (1, 0), (1, 4), (6, 9) and (0, 9) respectively. The arc BC is part of the curve with equation \(x^2 + (y - 10)^2 = 37\). The closed shape OABCD is formed, in turn, from the line segments OA and AB, the arc BC and the line segments CD and DO (see diagram). A funnel can be modelled by rotating OABCD by \(2\pi\) radians about the y-axis. \includegraphics{figure_9} Find the volume of the funnel according to the model. [3]

\includegraphics{figure_12} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the \(x\)-axis is rotated by \(2\pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\). Determine the exact volume of \(S\). [7]

In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac{1}{2}x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{2}\pi\). \includegraphics{figure_4} This region is rotated through \(2\pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated. [3]

Fig. 6 shows the region enclosed by part of the curve \(y = 2x^2\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(P(1, 2)\). \includegraphics{figure_1} The shaded region is rotated through \(360°\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed. [7]

A finite region is bounded by the curve with equation \(y = x + x^{-\frac{3}{2}}\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) This region is rotated through \(2\pi\) radians about the \(x\)-axis. Show that the volume generated is \(\pi\left(a\sqrt{2} + b\right)\), where \(a\) and \(b\) are rational numbers to be determined. [5 marks]

In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh\frac{1}{2}x^2\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360°\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm. [7]

A candlestick has base diameter \(8\) cm and height \(28\) cm, as shown in Figure \(9\). A model of the candlestick is shown in Figure \(10\), together with the equations that were used to create the model. \includegraphics{figure_7}

1. Show that the volume generated by rotating the shaded region (in Figure \(10\)) \(2\pi\) radians about the \(y\)-axis is \(\frac{112}{15}\pi\). [4]
2. Hence find the volume of metal needed to create the candlestick. [4]

\includegraphics{figure_9} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(ABCDEFGHA\) of the bottle with measurements taken by the student. The horizontal cross-section between \(CF\) and \(DE\) is a circle of diameter 8 cm and the horizontal cross-section between \(BG\) and \(AH\) is a circle of diameter 2 cm. The student thinks that the curve \(GF\) could be modelled as a curve with equation $$y = ax^2 + b \qquad 1 \leq x \leq 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Find the value of \(a\) and the value of \(b\) according to the model. [2]
2. Use the model to find the volume of water that the bottle can contain. [7]

In this question you must show detailed reasoning. \includegraphics{figure_12} The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{2 + t}}, \quad y = \ln(1 + t), \quad 2 \leq t < \infty$$ The point \(P\) on curve \(C\) has \(x\)-coordinate \(\frac{1}{2}\).

1. Find the exact \(y\)-coordinate of \(P\). [1]

The tangent to \(C\) at \(P\) meets the \(y\)-axis at point \(Y\).

1. Determine the exact coordinates of \(Y\). [4]

The curve \(C\) and the line segment \(PY\) are rotated \(2\pi\) radians about the \(y\)-axis.

1. Determine the exact volume of the solid generated. Give your answer in the form \(\pi(\ln p + q)\), where \(p\) and \(q\) are rational numbers. [8]

[You are given that the volume of a cone with radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2 h\)]

A finite region is bounded by the curve with equation \(y = x + x^{\frac{3}{2}}\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) This region is rotated through \(2\pi\) radians about the \(x\)-axis. Show that the volume generated is \(\pi\left(a\sqrt{2} + b\right)\), where \(a\) and \(b\) are rational numbers to be determined. [5 marks]

In this question you must show detailed reasoning. The diagram shows the curve with equation \(y = \frac{x + 3}{\sqrt{x^2 + 9}}\). \includegraphics{figure_8} The region R, shown shaded in the diagram, is bounded by the curve, the \(x\)-axis, the \(y\)-axis, and the line \(x = 4\).

1. Determine the area of R. Give your answer in the form \(p + \ln q\) where \(p\) and \(q\) are integers to be determined. [6]

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

1. Determine the volume of the solid of revolution formed. Give your answer in the form \(\pi\left(a + b\ln\left(\frac{c}{d}\right)\right)\) where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]

\includegraphics{figure_4} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the x-axis is rotated by \(2\pi\) radians around the x-axis to form a solid of revolution, S. Determine the exact volume of S. [7]