How can I decide I should use r or not?

Question

In my final exam, there were a divergence theorem question. Our 3-dimensional domain was a curve which has radius 4 and $y\le0$, $z\le0$

I have found $divF=4xy$ and I obtain this integral below.

$$ \iiint_D 4xydV$$

I have used cylinderical coordinates

$x=rcos\theta$ , $y=rsin\theta$, $z=z$

$x^2+y^2+z^2=16$ , $y\le0$, $z\le0$

$$\iiint_D 4xydV = 8\int_0^{\pi/2}cos\theta sin\theta d\theta \int_0^4r^3dr \int_0^ \sqrt{16-r^2}dz $$

I got zero because I should have written $x=4cos\theta$ , $y=4sin\theta$

Lecturer has said that we should take radius instead of $r$ variable in some situations. How can I decide, what are these situations?

Thanks in advance

Answer 1

I hope to have clearly understood your doubt.

In this case you are integrating over a solid domain thus r is variable.

You will take a specific values for r (e.g. r=4) when you are integrating over a cylindrical surface with a fixed radius.