<s>[INST] Use polar coordinates to evaluate function 
To find:

$$\iint_R\frac{1}{1+x^2+y^2}\,dA$$

where R is first quadrant bounded by $y=0$, $y=x$, $x^2+y^2=4$
  $$\iint_Rf(r\cos \theta,r\sin\theta)r\,dA$$
which is as far as I got -

$$\iint\frac{1}{1+r^2\cos^2\theta +r^2\sin^2\theta}\,dA$$
$$I=\iint \frac{1}{1+r^2}\,dr\,d\theta$$
$$I=\int\tan{^{-1}}r \,d\theta$$

Where to from here?

 [/INST] So
$$\iint_R\frac{1}{1+x^2+y^2}\,dA$$
Since $r^2 = x^2 + y^2$ and $dA = r \,dr \,d\theta$
$$\iint_R\frac{1}{1+r^2}r \,dr \,d\theta$$
Now, $R$ is the region bounded by $r = [0, 2]$ and $\theta = [0, \pi/4]$. So
$$\int_{0}^{\pi/4} \int_{0}^{2} \frac{r}{1+r^2} \,dr \,d\theta$$
$$\int_{0}^{\pi/4} \,d\theta \times \int_{0}^{2} \frac{r}{1+r^2} \,dr$$
$$\left( \frac{\pi}{4} \right) \left. \frac{1}{2} \ln(r^2+1)\right|_{r=0}^{2}$$
$$\left( \frac{\pi}{8} \right) \left[ \ln(5) - \ln(1)\right]$$
$$\left( \frac{\pi}{8} \right) \ln(5)$$
</s>