Volume with exponential or logarithmic functions

5. \begin{figure}[h] \end{figure} Diagram not drawn to scale The finite region \(S\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = \ln 4\) and the curve with equation $$y = \mathrm { e } ^ { x } + 2 \mathrm { e } ^ { - x } , \quad x \geqslant 0$$ The region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Use integration to find the exact value of the volume of the solid generated. Give your answer in its simplest form.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { e } ^ { 0.5 x } - 2\) The region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
Show that the volume of this solid can be written in the form \(a \ln 2 + b\), where \(a\) and \(b\) are constants to be found.

2. The diagram shows the curve with equation \(y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Find the volume of the solid formed when the shaded region is rotated through four right angles about the \(x\)-axis, giving your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are integers.

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.

5. \includegraphics[max width=\textwidth, alt={}, center]{825f6c7d-5399-4e7f-bacd-b7c0831aab06-1_408_858_1893_488} The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\) is rotated through four right angles about the \(x\)-axis. Find, in terms of \(\pi\) and e, the exact volume of the solid formed.

3 Fig. 4 shows the curve \(y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }\), and the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\). \begin{figure}[h] \end{figure}

1. Find the exact volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. 1. Complete the table of values, and use the trapezium rule with 4 strips to estimate the area of the shaded region.

      \(x\)	0	0.5	1	1.5	2
      \(y\)		1.9283	2.8964	4.5919

   2. The trapezium rule for \(\int _ { 0 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { 2 x } } \mathrm {~d} x\) with 8 and 16 strips gives 6.797 and 6.823, although not necessarily in that order. Without doing the calculations, say which result is which, explaining your reasoning.

5 Fig. 2 shows the curve \(y = \sqrt { } 1 + \mathrm { e } ^ { 2 x }\). \begin{figure}[h] \end{figure} The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of the solid of revolution produced is \(\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)\).

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

5 Fig. 4 shows the curve \(y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }\), and the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\). \begin{figure}[h] \end{figure}

1. Find the exact volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. 1. Complete the table of values, and use the trapezium rule with 4 strips to estimate the area of the shaded region.

      \(x\)	0	0.5	1	1.5	2
      \(y\)		1.9283	2.8964	4.5919

   2. The trapezium rule for \(\int _ { 0 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { 2 x } } \mathrm {~d} x\) with 8 and 16 strips gives 6.797 and 6.823, although not necessarily in that order. Without doing the calculations, say which result is which, explaining your reasoning.

9

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x - x ) = \ln x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{8492b214-aaac-4354-8649-e317bf7b3535-3_481_725_1064_751} In the diagram, \(C\) is the curve \(y = \ln x\). The region \(R\) is bounded by \(C\), the \(x\)-axis and the line \(x = \mathrm { e }\).
   (a) Find the exact volume of the solid of revolution formed by rotating \(R\) completely about the \(x\)-axis.
   (b) The region \(R\) is rotated completely about the \(y\)-axis. Explain why the volume of the solid of revolution formed is given by $$\pi \mathrm { e } ^ { 2 } - \pi \int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 y } \mathrm {~d} y ,$$ and find this volume.

4 Fig. 4 shows the curve \(y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }\), and the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\). \begin{figure}[h] \end{figure}

1. Find the exact volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. 1. Complete the table of values, and use the trapezium rule with 4 strips to estimate the area of the shaded region.

      \(x\)	0	0.5	1	1.5	2
      \(y\)		1.9283	2.8964	4.5919

   2. The trapezium rule for \(\int _ { 0 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { 2 x } } \mathrm {~d} x\) with 8 and 16 strips gives 6.797 and 6.823, although not necessarily in that order. Without doing the calculations, say which result is which, explaining your reasoning.

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.

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)

1. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Find the volume of the solid formed when the shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are integers.
□

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

1. Find the mean value of the function \(f\) for \(0 \leqslant x \leqslant 2\), giving your answer correct to three decimal places.
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.

7. (a) A curve \(C\) is defined by the equation \(y = \frac { 1 } { \sqrt { 16 - 6 x - x ^ { 2 } } }\) for \(- 3 \leqslant x \leqslant 1\).

1. Find the mean value of \(y = \frac { 1 } { \sqrt { 16 - 6 x - x ^ { 2 } } }\) between \(x = - 3\) and \(x = 1\).
2. The region \(R\) is bounded by the curve \(C\), the \(x\)-axis and the lines \(x = - 3\) and \(x = 1\). Find the volume of the solid generated when \(R\) is rotated through four right-angles about the \(x\)-axis.
   (b) Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \frac { - 8 \mathrm { e } ^ { - 2 x } } { 4 \mathrm { e } ^ { - 2 x } - 5 } \mathrm {~d} x$$ giving your answer correct to three decimal places.
   \section*{PLEASE DO NOT WRITE ON THIS PAGE}

1 In this question you must show detailed reasoning.

1. Show that \(\cosh ( 2 \ln 3 ) = \frac { 41 } { 9 }\). The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \sqrt { \operatorname { sinhx } }\), the \(x\)-axis and the line with equation \(x = 2 \ln 3\) (see diagram). The units of the axes are centimetres. \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-2_652_668_740_242} A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the shape of the solid formed by rotating \(R\) completely about the \(x\)-axis.
2. Determine, according to the model, the exact volume of one chocolate piece.

4 In this question you must show detailed reasoning. The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }\), the \(x\)-axis and the lines with equations \(x = \frac { 1 } { 2 }\) and \(x = 1\) (see diagram). The units of the axes are cm . \includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when \(R\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.

8 \includegraphics[max width=\textwidth, alt={}, center]{e792797e-6d20-4fc3-9733-43db10f764d7-4_677_1182_587_440} 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 \(S\).

2 In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac { 8 } { \sqrt { 16 + x ^ { 2 } } }\), the \(x\)-axis and the lines \(x = 0\) and \(x = 4\). Region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume generated. [4]

10. \begin{figure}[h] \end{figure} In Fig. 1, the curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x + \frac { 2 } { x ^ { 2 } } , \quad x > 0$$ 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.
[0pt] [P2 January 2002 Question 4]

5 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } - 9\). The graph cuts the coordinate axes at ( \(0 , a\) ) and ( \(b , 0\) ). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-004_817_908_479_550}

1. State the value of \(a\), and show that \(b = \ln 3\).
2. Show that \(y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81\).
3. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed, giving your answer in the form \(\pi ( p \ln 3 + q )\), where \(p\) and \(q\) are integers.
4. Sketch the curve with equation \(y = \left| \mathrm { e } ^ { 2 x } - 9 \right|\) for \(x \geqslant 0\).

5 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } - 9\). The graph cuts the coordinate axes at \(( 0 , a )\) and \(( b , 0 )\). \includegraphics[max width=\textwidth, alt={}, center]{908f530c-076d-47b1-90dd-38dbfe44f898-03_826_924_477_541}

1. State the value of \(a\), and show that \(b = \ln 3\).
2. Show that \(y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81\).
3. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed, giving your answer in the form \(\pi ( p \ln 3 + q )\), where \(p\) and \(q\) are integers.
4. Sketch the curve with equation \(y = \left| \mathrm { e } ^ { 2 x } - 9 \right|\) for \(x \geqslant 0\).

2 Fig. 2 shows the curve \(y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }\). \begin{figure}[h] \end{figure} The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of the solid of revolution produced is \(\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)\).