Double Integrals in Polar Coordinates

Try integrating over a disk in x and y: the boundary forces limits like y from −√(1−x²) to +√(1−x²). The algebra is brutal. But a disk is the most natural shape in the world — it just needs the right coordinates. In polar, a point is (r, θ): distance from the origin and angle. Suddenly the disk is a rectangle in disguise.