MathDB

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].

Problem Statement