Solving for Volume: Triple Integral of Wedge in Cylinder

[schaefera]

Homework Statement

Why does 2*Integral (r dz dr dt) for 0<z<rcos(t), 0<r<a, 0<t<pi/2 equal (2a^3)/3, when Integral (r dz dr dt) for 0<z<rcos(t), 0<r<a, 0<t<pi equal 0? All you are doing is using the fact that rcos(t) is an even function to make the limits easier, right?

Homework Equations

I'm practicing finding the volume of a known cross section with a wedge taken out from a cylinder at an angle of pi/4 radians; and I'm using the triple integral to check me work.

The Attempt at a Solution

My question more arose out of confusion than it being an assigned problem, but I can't quite figure it out!
Thanks for your help!

 

[Dick]

It's just because the integral of cos(t) over 0<t<pi/2 is 1. And the integral of cos(t) over 0<t<pi is 0. cos(t) is negative on (pi/2,pi).

 

[schaefera]

So to do this problem is it ok to take that 2 out and double the region of 0 to pi/2 because we want volume, which obviously can't be 0?

 

[Dick]

So to do this problem is it ok to take that 2 out and double the region of 0 to pi/2 because we want volume, which obviously can't be 0?

I have no idea. What problem are you actually trying to solve??