I am trying to find the volume bounded by the following surfaces.
$\ x^2 + y^2 + z^2 = 4$
$\ z = r^2$
I would like to set up an integral for this in Cartesian, Cylindrical, and Spherical coordinate systems. I am stuck on this because I don't quite know what to do with the r in the second surface when it comes to Cartesian coordinates.
Using $\ r^2 = x^2 + y^2 $(I thought this could only be used when going from Cartesian to Cylindrical, but I don't know what else to do with it) I got this far for Cartesian.
$\ \int_{-2}^2\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{x^2+y^2}^{\sqrt{4-x^2-y^2}} 1 dzdxdy $
But I don't think this is right as it yields a negative result.
You can slice it on disks along z axis, each disk is a cylinder with height equal to $dz$ and base is a circle with radius $R=\sqrt{x^2+y^2}$. Since $x^2+y^2=4-z^2$ you have $\int_{r^2}^2(4-z^2)\pi dz$