A curve has equation \(y=\mathrm{e}^{x}\) for \(\ln \frac{4}{3} \leqslant x \leqslant \ln \frac{12}{5}\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
(a) Use the substitution \(u=\mathrm{e}^{x}\) to show that
\(A=2 \pi \int_{\frac{4}{3}}^{\frac{12}{5}} \sqrt{1+u^{2}} \mathrm{~d} u\)
(b) Use the substitution \(u=\sinh v\) to show that
\(A=\pi\left(\frac{904}{225}+\ln \frac{5}{3}\right)\)