Double Integrals in Polar Coordinates

The central idea

Under $x=r\cos\theta$, $y=r\sin\theta$ with $r\ge0$, area transforms as $dA=r\,dr\,d\theta$. Thus $\iint_Df(x,y)dA=\iint_{D^*}f(r\cos\theta,r\sin\theta)r\,dr\,d\theta$. The extra $r$ is the absolute Jacobian determinant and must be included.

Definitions, hypotheses, and notation

Regions offset from the origin can require radial bounds from solving a polar equation, such as $r=2\cos\theta$ for a shifted circle. Negative radial bounds are usually avoided in change-of-variables integrals by choosing angles and $r\ge0$ that cover the region once. The coordinate map is singular at $r=0$, but that single point has zero area and causes no problem for ordinary integrable functions.

Order $drd\theta$ describes rays; order $d\theta dr$ describes arcs at fixed radius and may require splitting a region differently. Choose bounds from geometry rather than copying a standard disk pattern.

Conceptual meaning

A small polar cell is approximately a sector rectangle with radial thickness $dr$ and arc length $r d\theta$, so its area is $rdrd\theta$. Polar coordinates fit disks, annuli, sectors, and radial integrands naturally.

A dependable method and decision rule

1. Sketch the planar region and choose an angular interval without repeated coverage.
2. Write radial bounds along each ray.
3. Convert every occurrence of $x,y$, especially $x^2+y^2$.
4. Multiply the transformed integrand by the Jacobian factor $r$.
5. Integrate and compare with area or symmetry estimates.

Fully worked example

Graphical or geometric meaning

Concentric rings at radius $r$ have circumference $2\pi r$, so equal radial thicknesses contain more area farther from the origin. The Jacobian factor expresses that growth.

Common mistakes and why they fail

Verification and reasonableness checks

• Integrate $1$ and recover the known region area.
• Verify radial bounds are nonnegative and ordered.
• Check units and rough size using the integrand's extrema.

The Jacobian is part of the area

The polar area element is $dA=r\,dr\,d\theta$, not just $dr\,d\theta$. The factor $r$ records how a small angular wedge widens away from the origin. Convert both the integrand and the region using $x=r\cos\theta$, $y=r\sin\theta$, and $x^2+y^2=r^2$. Draw the angular sweep and radial bounds; a careless $0\le\theta\le2\pi$ may cover a symmetric piece more times than intended. Standard polar integration takes $r\ge0$, avoiding the coordinate duplication caused by negative radii. Test the setup with integrand one: the result should match the familiar area of the disk, annulus, or sector before tackling a more complicated density.

Practice

1. Find area of the disk $r\le3$ by double integration.
2. Integrate $r^2$ over the unit disk.
3. What is the polar Jacobian?

Connections and next steps

• Double Integrals
• Polar Coordinates in Calculus
• Triple Integrals in Cylindrical Coordinates