Double Integral with Polar Cordinates

Question

$$ \int \:\int _r^{ }\:\:x^2+y^2\:dxdy $$

The question is asking me to change these cartesian coordinates to polar coordinates which I have been unable to do, and then it provides me with R is bound in the first region by $ y = 0$ and $y = x$ and the circle $x^2 + y^2 = 4$.

Can someone please help me understand this

Answer 1

Your domain in polar coordinates is

$$R=\{(r,\theta)\,|\,0<\theta<\pi/4\,,\,0<r<2\,\}$$ and then your integral reads

$$\int\int_R x^2+y^2 dx\,dy=\int_0^{\pi/4}\int_0^2 r^3\,dr\,d\theta=\pi$$

Answer 2

Area element $ dx\cdot dy$ changes to $ r d \theta \cdot dr\,$ $$\int_0^2 \int_{\pi/4} ^{\pi/2} r^2 r dr\, d\theta $$

Limits are separable, $r$ is not a function of $\theta$

$$ \dfrac{r^4}{4}\big|_0^2 \cdot \theta \big|_{\pi/4}^{\pi/2}=\,\pi$$