Volume with exponential 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.

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.

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.

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)

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 \(\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)\).

8 Find the exact volume of the solid of revolution generated by rotating the graph of \(y = 3 \mathrm { e } ^ { x }\) between \(x = 0\) and \(x = 2\) through \(360 ^ { \circ }\) about the \(x\)-axis.