Calculus AB...I really need help
posted by Laura .
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.
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!
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)]