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) = \frac1{\sqrt2} \implies \theta = \frac\pi4 \text { or } \theta = \frac{3\pi}4`

Then the same integral in polar coordinates is

`\displaystyle \int_(-1)^1 \int_1^(√(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}`