Question

The region in the first quadrant enclosed by the coordinates axes, the line x=pi, and the curve y= cos(cosx) is rotated about the x-axis. What is the volume of the solid generated.

Answer 1

v = ∫πy^2 dx [0,π]
= ∫πcos^2(cosx) dx [0,π]

that is not something you can evaluate using elementary functions. wolframalpha can do it, but it's done numerically, fer shure!

Answer 2

Using cos(2 x) = 2 cos^2(x) - 1 and the definition of the Bessel function of zeroth order:

J0(x) = 1/pi Integral from zero to pi of cos[x cos(t)] dt,

you find that the volume is given by:

pi^2/2 [1 + J0(2)]