Set up an integral in cylindrical coordinates to evaluate $\iiint_{E} x y d V$ where $E$ is the region enclosed by the cone $z=2-\sqrt{x^{2}+y^{2}}$, the cylinder $x^{2}+y^{2}=1$, and the $x y$ plane.
My try:
The region $E$ is shown below:
So the triple integral in cylindrical coordinates is:
$$\int_{\theta=0}^{2 \pi} \int_{r=0}^{1} \int_{z=0}^{2-r}(r \cos \theta)(r \sin \theta) r d z d r d \theta$$
Is this set up correct?