Question

Calculate the area of ​​the region R in the xy plane enclosed in the circumference

x² + y²= 4, to the right of line x = 1 and above line x =√3y.

Answer 1

$x=rcos\theta,y=rsin\theta$

$r=2$

For upper limit:

$2cos\theta=1$

$\theta=\pi/3$

For lower limit:

$cos\theta=\sqrt{3}sin\theta$

$\theta=\pi/6$

$S=1/2\int r^2d\theta=1/2\int^{\pi/3}_{\pi/6}4d\theta=2(\pi/3-\pi/6)=\pi/3$