find the volume of the region bounded by the paraboloid $z=x^2+y^2$ and the cylinder $x^2+y^2=9$ My attempt:
$\int_{x=-3}^3 \int_{y=-3}^3 x^2+y^2 dy dx$
$\int_{-3}^3 6x^2 + 18 dx$
$= 216?$

The limits of your solution actually define the squared cylinder $[-3,3]\times[-3,3]$. You could either try by changing the limits to $-\sqrt{9 - x^2} \le y \le \sqrt{9 - x^2} $ and $-3 \le x \le 3$ or by changing your coordinate system to $x = r\cos\theta$, $y = r sin\theta$
$$
\int\int_{x^2 + y^2 < 9} dxdy\; (x^2+y^2) = \int\int_{r<3} dr r\; (r^2) = 2\pi \int_0^3dr\; r^3 = \frac{81}{2}\pi
$$
