Find the volume below the graph of the paraboloid $z=3-x^2-y^2$. I have to calculate this integral without using polar coordinates.
The integration's limits are not specified so I've assumed that the volume is between the $xy$ plane and the graph. If so, I end up having $0 \leq z \leq 3-x^2-y^2$ as a domain of integration but I don't know how to use it in order to write and calculate the integral.
The bounds on $x$ and $y$ are $-\sqrt3$ to $\sqrt3$, as the intersection of the paraboloid with the $xy$-plane is the circle $x^2+y^2=3$. This gives us the integral $$\int_{-\sqrt3}^{\sqrt3}\int_{-\sqrt{3-x^2}}^{\sqrt{3-x^2}}\int_0^{3-x^2-y^2}1\,dz\,dy\,dx$$ which I leave to you to evaluate.