why is this volume $0$? (double integral, polar coordinates) I got that $$\iint x^2-y^2 dA=0$$ over the region $x^2+y^2=1$ (I used polar coordinates). But why is that? How can a volume be 0?

Do not forget that an integral sums up function values!
That double integral is telling you to sum up all the function values of $x^2 - y^2$ over the unit circle.
To get $0$ here means that either the function does not exist in that region OR it's perfectly symmetrical over it. Knowing that $x^2 - y^2$ is a hyperbolic parabola, it definitely exists in this region, and getting a $0$ answer means that half of the function must be below the plane, while the other half is above it, making it the summation of those function values total to $0$.
So the volume here isn't actually $0$, but the integral is. If you wanted to know the actual geometric volume, you would have to split up the surface to the part that is above the plane, and then multiply by $2$.
Sorry for the lengthy response, but I hope that clears up how you get "$0$" volume.
