MathDB

For a constant

a

, denote

C(a)

the part

x\geq 1

of the curve

y=\sqrt{x^2-1}+\frac{a}{x}

.
(1) Find the maximum value

a_0

a

such that

C(a)

is contained to lower part of

y=x

, or

y<x

.
(2) For

0<\theta <\frac{\pi}{2}

, find the volume

V(\theta)

of the solid

V

obtained by revoloving the figure bounded by

C(a_0)

and three lines

y=x,\ x=1,\ x=\frac{1}{\cos \theta}

about the

x

-axis.
(3) Find

\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)

.
1992 Tokyo University entrance exam/Science, 2nd exam

Problem Statement