Polar coordinates

The pair \((r,\theta)\) is related to \((x,y)\) by \(x=r\cos\theta\), \(y=r\sin\theta\), and conversely \(r^2=x^2+y^2\), \(\tan\theta=y/x\). The representation is not unique: \(\theta\) and \(\theta+2\pi\) give the same point, and the origin has \(r=0\) with any angle. When recovering \(\theta\) you must watch the quadrant, because \(\arctan(y/x)\) only returns angles between \(-\tfrac{\pi}{2}\) and \(\tfrac{\pi}{2}\).

A «polar rectangle» is \(\{a\le r\le b,\ \alpha\le\theta\le\beta\}\): a ring or a sector. There \(\iint_D f\,dA=\int_\alpha^\beta\!\int_a^b f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta\). The factor \(r\) is the Jacobian of the change of variables: a cell at distance \(r\) measures \(dr\) in the radial direction and \(r\,d\theta\) along the arc, so its area is not \(dr\,d\theta\).

Guiding example: \(\iint_D (x^2+y^2)\,dA\) over the unit disc is \(\int_0^{2\pi}\!\int_0^1 r^3\,dr\,d\theta=\tfrac{\pi}{2}\), whereas in Cartesian form it requires integrating \(\sqrt{1-x^2}\) and powers of it. With the same method, \(\iint_{\mathbb{R}^2}e^{-(x^2+y^2)}dA=\pi\), hence \(\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt\pi\). If the region is \(g_1(\theta)\le r\le g_2(\theta)\), the inner limits depend on \(\theta\), just as in a general region.

Typical mistakes: forgetting the factor \(r\), leaving \(x\) or \(y\) unsubstituted in the integrand, and choosing polar coordinates when neither the region nor the integrand is circular. This change of variables is the planar case of cylindrical coordinates, and both are examples of the general change of variables with the Jacobian.