Circular segment using polar coordinates

Question

Imagine, you'd like to integrate $f(x,y)=x^2-y$ over this region:

How would you parameterize this circular segment? My try: $$\Phi : [1,2]\times [-\pi /3, \pi /3]\to \mathbb{R} \, (r,\varphi)\mapsto (x,y)=(r\cos(\varphi),r\sin(\varphi))$$ is this correct?

Answer 1

HINT

The integral you are interested in is given by: \begin{align*} I = \int_{\arctan(-\sqrt{3})}^{\arctan(\sqrt{3})}\int_{\sec(\theta)}^{2}rf(r\cos(\theta),r\sin(\theta))\mathrm{d}r\mathrm{d}\theta \end{align*}

Can you take it from here?

Answer 2

By symmetry, the $y$ term doesn't contribute, so we just want$$2\int_0^{\pi/3}d\theta\cos^2\theta\int_{\sec\theta}^2r^3dr=\int_0^{\pi/3}d\theta(4+4\cos2\theta-\tfrac12\sec^2\theta)=4\pi/3+\sqrt{3}/2.$$