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!
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.
2 answers
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)]
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)]