Integrate using polar $\int_{-1}^2 \int_0^{\sqrt{4-x^2}} x^2 + y^2 dy dx$

Question

If it would be whole upper half of the disk then it would be easy, but I don't know how to integrate cutted disk using polar coordinates.

$$\int_{-1}^2 \int_0^{\sqrt{4-x^2}} x^2 + y^2 dy dx$$

Answer 1

Divide the polar integral into two part as

$$\int_0^{\frac{2\pi}{3} }\int_0^2 r^3drd\theta + \int_{\frac{2\pi}{3}}^\pi\int_0^{-\frac1{\cos\theta}} r^3drd\theta $$