$A=\{(x,y)\in\mathbb{R^2}:x^2+y^2\le1, 0\le x\}$
Calculate $\iint_A\frac{1}{1+x^2+y^2}dxdy$
I'm doing that problem from an exam, and the solution is the following:
Use polar coordinates, $x=r\cos(\theta),y=r\sin(\theta).$ Then we use a change of variables $u=1+r^2$ and we get:
$\iint_A\frac{1}{1+x^2+y^2}\,dx\,dy=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{1}\frac{1}{1+r^2}\,dr\,d\theta=\frac{\pi}{2}\int_{1}^{2}\frac{1}{u}du=\frac{\pi \log(2)}{2}$
Yet, I did the following:
$\iint_A\frac{1}{1+x^2+y^2}\,dx\,dy=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{1}\frac{1}{1+r^2}\,dr\,d\theta=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\pi}{4}\,d\theta=\frac{\pi^2}{4}$
Where is the error which clearly I'm not seeing?