Triple Integral using Cylindrical coordinates

[elizabeth_b]

Hi! These triple integrals are confusing me. I don't know how to type this out in math symbols so I'm hoping you'll be able to read my english. The problem is the triple integral over "S'' of "x dV" where S is the region bounded above the plane z=x+y, below by the xy-plane, and on the sides by the cylinder r=1.
So this is what I have so far, but I'm not sure if its right
z=x+y -----> z=rcos? + rsin?
r=1
x=rcos?
r ranges from 0 to 1
? ranges from ?
Right now my integral looks like this:
the first intergrand ranges from ?
the 2nd ranges from 0 to 1
the 3rd ranges from 0 to rcos?+rsin?
then i integrate (r^2cos? dz dr d?) using the bounds above...
Except I can't figure out what ? ranges are. And I'm not even sure if the other stuff is right! I would so appreciate help!

 

[galactus]

Theta would range over 0 to 2Pi.

\(\displaystyle \int_{0}^{2\pi}\int_{0}^{1}\int_{0}^{r(cos{\theta}+sin{\theta})}r^{2}cos{\theta}dzdrd{\theta}\)

 

[elizabeth_b]

Okay... Well I have the integral down to:
(1/4)*integral of [cos^2(theta)+sin(theta)cos(theta) dtheta] ranging from 0 to 2pi.
How do you integrate that?

 

[Deleted member 4993]

Okay... Well I have the integral down to:
(1/4)*integral of [cos^2(theta)+sin(theta)cos(theta) dtheta] ranging from 0 to 2pi.
How do you integrate that?

\(\displaystyle cos^2\theta\, = \, \frac{1\,+\, cos(2\theta)}{2}\)

 

[elizabeth_b]

Thanks!