Change variables of triple integrals

Question

I'm trying to solve this problem, the following is what I have done:

Can anyone please tell me how to change it to (θ,φ,z)? Thanks!

Answer 1

Since the region $V$ has a cylindrical symmetry, you may want to use cylindrical coordinates $(r,\phi,z)$.

$V$ then corresponds to the region $\sqrt{1+r^2} < z < \sqrt{4-r^2}$. Notice that the surfaces meet at $r = \sqrt{\frac{3}{2}}$, so this will be our upper bound for $r$.

Furthermore, we need to use the Jacobian $r$, since we want to transform from cartesian $(x,y,z)$ to cylindrical coordinates $(r,\phi,z)$.

In your case this becomes $$ \iiint_V |yz|\operatorname{d}V = \int_{r=0}^{\sqrt{\frac{3}{2}}} \int_{\phi=0}^{2\pi} \int_{z=\sqrt{1+r^2}}^{\sqrt{4-r^2}} |(r\sin\phi)z| r \; \text{d}z \text{d}\phi\text{d}r.$$