Here's the question $$\iiint (x-2) \,dx\,dy\,dz.$$ I am asked to evaluate this integral over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :1\leq x^2+y^2+z^2 \leq9,x\leq z,z \geq 0\right \}.$$ I tried to use the spherical coordinates find the solution :
$\begin{cases} z=\rho cos\phi \\ x=sin\phi cos\theta \\ z=sin\phi sin \theta \end{cases}$
I obtained that:
- $ 0 \leq \rho \leq 1 $
- $ 0 \leq \phi \leq \frac{\pi}{2}$
- $ sin \phi * cos \theta\leq cos \phi $
How can i obtain the value of $\theta$ or $\phi$ from the third inequality? What can I do or what have I done wrong up until now?
Any support for this question would be appreciated.
Not an answer but "support for this question" (so please don't downvote): a figure to help visualization.
It is clear that the OP's inequality on radius is incorrect. Instead: $1 \leq \rho \leq 3$