Double Integrals in Polar Coordinates

Summary

When a region or integrand is circularly symmetric, switch to polar coordinates with $x = r \cos θ$ , $y = r \sin θ$ , and area element $d A = r d r d θ$ . The extra factor $r$ is the absolute value of the Jacobian.

Prerequisites

Double Integrals, Polar Coordinates

Formula

\iint_{R} f (x, y) d A = \iint_{R^{'}} f (r \cos θ, r \sin θ) r d r d θ .

Conditions / Assumptions

$f$ continuous on the compact region; $r \geq 0$ .
Describe $R^{'}$ with constant or simple bounds in $(r, θ)$ when possible (polar rectangles, sectors, annuli).

Worked Example

Disk area

Area of $x^{2} + y^{2} \leq R^{2}$ :

A = \int_{0}^{2 π} \int_{0}^{R} r d r d θ = \int_{0}^{2 π} \frac{R^{2}}{2} d θ = π R^{2} .

(Not $2 π R^{2}$ .)

Integrand $x^{2} + y^{2}$ on the disk

\iint_{x^{2} + y^{2} \leq R^{2}} (x^{2} + y^{2}) d A = \int_{0}^{2 π} \int_{0}^{R} r^{2} \cdot r d r d θ = \int_{0}^{2 π} \frac{R^{4}}{4} d θ = \frac{π R^{4}}{2} .

Common Mistakes

Forgetting the Jacobian factor $r$ .
Computing disk area as $2 π R^{2}$ by integrating $r$ incorrectly or double-counting.
Using Cartesian limits after substituting polar expressions.

Connections

Polar Coordinates, Double Integrals over General Regions, cylindrical/spherical analogs in 3D

References

Polar double integrals are in OpenStax Calculus Volume 3.^[1]

OpenStax, Calculus Volume 3, Section 5.3, https://openstax.org/details/books/calculus-volume-3 ↩︎