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