Putnam 1940 B2

A cylindrical hole of radius $r$ is bored through a cylinder of radiues $R$ ( $r\leq R$ ) so that the axes intersect at right angles. i) Show that the area of the larger cylinder which is inside the smaller one can be expressed in the form $$S=8r^2\int_{0}^{1} \frac{1-v^{2}}{\sqrt{(1-v^2)(1-m^2 v^2)}} dv,\; \text{where} \; m=\frac{r}{R}.$$ ii) If $K=\int_{0}^{1} \frac{1}{\sqrt{(1-v^2)(1-m^2 v^2)}} dv$ and $E=\int_{0}^{1} \sqrt{\frac{1-m^2 v^2}{1-v^2 }}dv$ . show that $$S=8[R^2 E - (R^2 - r^2 )K].$$