Expert: Paul Klarreich - 11/25/2005

Question
I have a problem that I am having a hard time solving. I need to find the volume of the solid obtained when the region bounded by y=cos(x), x=0, x=pie/2, and y=0 is revolved around the x-axis. I figured the best way to solve this was the disk method, but I'm having a hard time integrating (cos(x))^2. Can you possibly help me?

Thanks!!
Emily

Answer
Hi, Emily,

Assuming you only need the integral of cos^2(x),there is a standard 'trick' for it.  It is called the 'half-angle' formula even though it doesn't seem to have any half-angles in it.
          1 + cos(2x)
cos^2(x) = -----------
              2

You shouldn't have any trouble integrating that.  If you are wondering where it comes from, recall these formulas:

cos(2x) = 2 cos^2(x) - 1, which you solve for cos^2(x) to get the above formula, and

sin(2x) = 1 - 2 sin^2(x), which you solve for sin^2(x) to get the other formula:
          1 - cos(2x)
sin^2(x) = ------------
               2

which you will need when you have to integrate sin^2(x)