Volume of revolution with substitution

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1. Find \(\int _ { 1 } ^ { \infty } \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).
   \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-16_499_689_1322_726} The diagram shows the curve with equation \(y = \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Find the volume of revolution.
   The normal to the curve at the point \(( 1,1 )\) crosses the \(y\)-axis at the point \(A\).
3. Find the \(y\)-coordinate of \(A\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1. (a) Using the substitution \(u = 4 x + 2 \sin 2 x\), or otherwise, show that

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 4 x + 2 \sin 2 x } \cos ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 8 } \left( \mathrm { e } ^ { 2 \pi } - 1 \right)$$ Figure 3 The curve shown in Figure 3, has equation $$y = 6 \mathrm { e } ^ { 2 x + \sin 2 x } \cos x$$ The region \(R\), shown shaded in Figure 3, is bounded by the positive \(x\)-axis, the positive \(y\)-axis and the curve. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
(b) Use the answer to part (a) to find the volume of the solid formed, giving the answer in simplest form.