# Multi-Variable Calc: Using Polar Coordinates to Find Double Integrals

Use polar coordinates to find the volume $V$ of the solid region $T$.

$T$ lies below the paraboloid $z = 9x^2 + 9y^2$, above the $xy$-plane, and inside the cylinder $x^2 + y^2 = 2y$.

I know I have to use a double integral but I do not know how to set it up.

Use polar variable changing \begin{cases} x=r\cos\theta,\\ y=r\sin\theta. \end{cases} with these changing we find $0\leq\theta\leq\pi$, and $x^2+y^2=2y$ gives $r^2=2r\sin\theta$ or $r=2\sin\theta$ and the volume below of function surface $$T:\,\,z=f(x,y)=9(x^2+y^2)=9r^2$$ is $${\bf V}=\int _{\theta_1 }^{\theta_2 }\int _{r_1}^{r_2 }z\,\color{blue}{r}\,drd\theta=\int _{0 }^{ \pi }\int _0^{2 \sin\theta}9 r^2\,\color{blue}{r}\,drd\theta=\color{blue}{\dfrac{27\pi}{2}}$$ where $\color{blue}{r}$ is changing in variable factor (called Jacobean)