Question

Volume of the area bounded by y=sin(x), x-axis, x is greater than or equal to 0 but less than or equal to pi, revolved about x=3pi/2.

Answer 1

Height = sin x
radius = (3 pi/2 -x) = r so dr=-dx
d volume = sin x (2 pi r (-dr))
= 2 pi sin x (3pi/2-x)dx
=2 pi(3pi/2) sin x dx - 2pixsin x dx
= -3 pi^2 cos x - 2 pi [sinx-xcosx]

= [-3 pi^2+2pix]cos x -2 pi sin x
at pi - at 0
sin is 0 at both so

cos pi = -1 so
- [-3 pi^2+2pi^2]at pi
and cos 0 = 1 so
-3 pi^2 at 0
in the end
6 pi^2 -2 pi^2
= 4 pi^2
check my arithmetic!!!