Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$

Question

Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$

Not sure how to proceed

Answer 1

You want to calculate the integral of the domain bounded by the surfaces:

The plane $S_1:=\{(x,y,z):z=0 \}$, $\quad$the infinite cylinder $S_2:=\{(x,y,z):x^2+y^2\leq2 \}$, $\quad$ and the parabolioid $S_3:=\{(x,y,z):z=x^2+y^2 \}$

This gives you a bounded set $D$ between all these surfaces and you want to calculate $\int_D1 dV$. More specifically write it as a some recurent integral which you can calculate easily. In this case one can see that $z$ ranges between $0$ and $2$, and for each $z$ there is a disc or radius $z$ centered at the $z$ axis. You can then define $D_z:=\{ (x,y): x^2+y^2 \}$, and write by Fubini:

$\int_D1 dV= \int_{z=0}^2 \Big( \int_{D_z}1dxdy \Big)dz$

Which I think is much easier to calculate.

Answer 2

Either the cylinder and the "cap" (and the base plane) are symmetrical for a rotation around the $z$ axis.
You can thus take a cross section with $y=0$, and $0 \le x$, and rotate it by $2\pi$. The cross section is delimited by a parabola, so it is easy to integrate by shell method.