Triple integral - how to find upper and lower bound

Question

I've got this integral.

$\int_RxyzdV$

on the domain

$R=\{(x,y,z):x^2+y^2+z^2\leq 1$ and $y\geq0$ and $z\geq 0\}$

I think I should convert it to spherical coordinates integration which I did, but then I don't know where I should integrate, I mean, the upper and lower bound. I've tried:

$0\leq\rho\leq 1$

$0\leq\phi\leq\pi/2$

$0\leq\theta\leq \pi/2$

But I'm not sure if its correct. Any help?

Thank you.

Answer 1

Hint: You have accounted for half of the volume you want.

Revisit your angular bounds. To cover all angles, one angular measure goes from $0$ to $2\pi$ (or $-\pi$ to $\pi$) and the other goes from $-\pi/2$ to $\pi/2$.

By limiting to $+y$ and $+z$ you'd be covering a quarter of all angles. Right now you're covering an eighth of all angles.

Spoiler answer:

$-\pi/2 \leq \phi \leq \pi/2$