Volume of the body

Question

Find the volume of body bounded with:$(z-2)^2\le x^2+y^2, z\ge 1,z-14+x^2+y^2\le 0$. So we should have three volumes here. For first one(inside the paraboloid and on the cone) we have $V_1=\int\limits_{0}^{2\pi}d\phi\int\limits_{0}^{1}rdr\int\limits_{2-r}^{2+r}dz$. And I am not sure for the other two volumes, one should be from plane $z=1$ to paraboloid(with projection in OXY plane $9\le x^2+y^2\le 13$). But what is the third one(I mean I see one from the left, but it should be symmetric to this one).

Answer 1

The integral can be split between $1 \leq z \leq 2$ and $2 \leq z \leq 5$

$\displaystyle \int_0^{2\pi} \int_1^2 \int_{2-z}^{\sqrt{14-z}} r \ dr \ dz \ d\theta \ + \int_0^{2\pi} \int_2^5 \int_{z-2}^{\sqrt{14-z}} r \ dr \ dz \ d\theta$