So my task is to evaluate this triple integral:
SSS yz DV, over the region E: above z = 0, below z = y, and inside the cylinder x^2 + y^2 = 4.
I thought it'd be best to use cylindrical coordinates. Thus:
0 <= z <= rsinθ
0 <= r <= 2
0 <= θ <= 2π
But I keep getting 0. The answer is actually 64/15.
Here are the steps I took to compute the integral:
= SSS r^2 * sinθ * z dz dr dθ
= SS r^2 * sinθ * [(z^2)/2] from 0 to rsinθ .. dr dθ
= SS (r^4 * (sinθ)^3)/2 dr dθ
= S ((sinθ)^3)/2 * [(r^5)/5] from 0 to 2 dθ
= (16/5) S ((sinθ)^3) dθ
use the trig identity to split it up (sinθ)^2 = 1 - (cosθ)^2, the use substitution... etc
the final result:
(16/5) * [((cosθ)^3)/3 - cosθ] from 0 to 2π
This is obviously zero, since cos(0) and cos(2π) produce the same result. What am I doing wrong?