Question

Write an integral in polar coordinates equivalent to the following:​

Answer 1

The region is a segment of a circle with radius √2 lying above the line $y=1$. In polar coordinates, this line has equation

$y = r\sin(\theta) = 1 \implies r = \csc(\theta)$

and the circle has equation

$x^2+y^2 = r^2 = 2 \implies r=\sqrt2$

The two curves meet when

$\csc(\theta) = \sqrt2 \implies \sin(\theta) = \dfrac1{\sqrt2} \implies \theta = \dfrac\pi4 \text { or } \theta = \dfrac{3\pi}4$

Then the same integral in polar coordinates is

$\displaystyle \int_{-1}^1 \int_1^{\sqrt{2-x^2}} xy \, dy \, dx = \boxed{\int_{\pi/4}^{3\pi/4} \int_{\csc(\theta)}^{\sqrt2} r^3 \cos(\theta) \sin(\theta) \, dr \, d\theta}$