Compute $$I=\iiint_Kx^2+y^2 \ dxdydz,$$
where $K=\{x^2+y^2\leq1, \quad 0\leq z\leq 1+\sqrt{1-x^2-y^2}\}.$
It's easy to see that $K$ is just a cylinder with height and radius $1$ in the $z-$direction and on top of it is a semishphere with radius $1$ and center at $(0,0,1).$ Let $D=\{x^2+y^2\leq 1\},$ we have then
$$I=\iint_D(x^2+y^2)\left(1+\sqrt{1-(x^2+y^2)}\right) \ dxdy.$$
Going over to polar coordinates I get
$$I=\iint_Er^2(1+\sqrt{1-r^2})\cdot r\ drd\theta=2\pi\int_0^1r^3 \ dr+2\pi\int_0^1r^3\sqrt{1-r^2}\ dr.$$
The first integral is easily dealt with. I wan't to find the primitive to $f(r)=r^3\sqrt{1-r^2},$ I know it is possible by repeated partial integration but I feel it's quite tedious and makes room for errors. Is there a neat way to treat integrands like these? If the powe inside the squareroot was greater than the power of the $r$ outside of it, a substitution would do it.