Given in my homework I have to compute (by hand) $$\iint\limits_{x^2+y^2\leq 1}(x^2+y^2)\,\mathrm dx\,\,\mathrm dy.$$ My solution so far: Let $f(x,y)=x^2+y^2$ and $K=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq 1\}$. With the transformation of polar coordinates $\varphi:(r,\phi)\mapsto(r\cdot\cos\phi,r\cdot\sin\phi)=(x,y)$ and the determinant of the Jacobian matrix $|\det D_\varphi|=r$ we can rewrite the set as $K=\{(r\cdot\cos\phi,r\cdot\sin\phi):r\in[0,1],\phi\in[0,2\pi]\}$ and our integrand too as $f(x,y)=x^2+y^2=r^2$. $$\iint\limits_K f(x,y)\,\mathrm dx\,\mathrm dy=\int\limits_0^1\int\limits_0^{2\pi}r^2\cdot |\det D_\varphi|\,\mathrm d\phi\,\mathrm dr=2\pi\int\limits_0^1r^3\,\mathrm dr=\left.2\pi\cdot\frac{1}{4}r^4\right|_0^1=\frac{\pi}{2}.$$
I now want to check the result with a CAS. I am pretty new to Mathematica so I just didn't found any clue on how to enter such an integral for computation. Any help out there?