Evaluating the integral of $f(x,y,z) = \frac{y}{\sqrt{z}}$ on $y \geq 0$ and $0 \leq z \leq x^2$ and $(x-2)^2+y^2 \leq 4$

Question

I am asked to evaluate the integral of $f(x,y,z) = \frac{y}{\sqrt{z}}$ on

$$ y \geq 0\\ 0 \leq z \leq x^2\\ (x-2)^2+y^2 \leq 4 $$

What I have so far (and it seems a little off) is

$$ \int_{0}^{\pi} \int_{0}^{4cos(\theta)} \int_{0}^{r^2cos^2(\theta)} \frac{r sin(\theta)}{\sqrt{z}} r \ dzdrd\theta $$

What am I going wrong?

Thank you.

ANSWER: $\frac{64}{3}$

Answer 1

I think the bounds on the very first integral should be changed from $0$ to $\dfrac{\pi}{2}$. Since we are looking at the circle $r=4cos(\theta)$ and only want the upper half of this circle, we should have the $\theta\in[0,\pi/2]$

Answer 2

Your limits for theta should be from o to $\frac{\pi}{2}$ as you have to take only in 1st quadrant because of y $\geq$ 0