Change of variables cylindrical coordinates

[manjum423]

Homework Statement

Let S be the part of the cylinder of radius 9 centered about z-axis and bounded
by y >= 0; z = -17; z = 17. Evaluate
$\iint xy^2z^2$

Homework Equations

The Attempt at a Solution

So I use the equation $x^2 + y^2 \leq 9$, meaning that r goes from 0 to 3
Since $y \geq 0$, θ goes from 0 to ∏
So the integral looks like this:
$\int_0^∏ \int_0^3 \int_{-17}^{17} (rcosθ)(rsinθ)^2 z^2 r dzdrdθ$
And I get:
$\int_0^∏ \int_0^3 \int_{-17}^{17} r^4 cosθsin^2θ z^2 dzdrdθ$
I'm having trouble evaluating this integral because for the $\int_0^∏ cosθsin^2θ$part. I get 0 (When u=sinθ by u-substitution and you get $(1/3)sin^3θ$ from 0 to ∏)
Basically I would like to know if my limits and my setup is correct, and if anyone can help me out with a solution I would be grateful.

 

[Vorde]

Well, firstly, the range of the dr-integral should be 0 to 9 not 0 to 9 because we're dealing with the cylinder of radius 9, not the equation x^2+y^2 <= 9.

But that won't affect the problem, I just did this myself and I also got 0. Unless someone else can point out a mistake we both made, I think that's the answer.

Edit: That makes sense, I think, because the equation w=xy^2z^2 is symmetric across the y axis.