Problem

(a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that

$$2\sinh^2A=\cosh 2A-1.$$

(b) A curve has equation $y=x^2$, for $0\le x\le\frac{2}{3}$. The area of the surface generated when the curve is rotated through $2\pi$ radians about the $x$-axis is denoted by $S$.

Use the substitution $x=\frac{1}{2}\sinh u$ to show that

$$S=\frac{1}{32}\pi\left(\frac{820}{81}-\ln 3\right).$$