---
title: "Double Integrals in Polar Coordinates"
description: "Integrate over circular regions by switching to polar coordinates, where the area element carries an extra factor of r."
canonical: https://lightmysky.com/learn/mathematics/double-integrals-in-polar-coordinates-mt_93e-KSzQMT
source: https://lightmysky.com/learn/mathematics/double-integrals-in-polar-coordinates-mt_93e-KSzQMT.md
retrieved: 2026-09-12
---

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# Double Integrals in Polar Coordinates

Integrate over circular regions by switching to polar coordinates, where the area element carries an extra factor of r.

Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/double-integrals-in-polar-coordinates-mt_93e-KSzQMT

## Ready when they can

- Convert a region and an integrand to polar form
- Explain where the factor of r in the area element comes from
- Evaluate an integral over a disc or an annulus

## Lesson: Round regions want radius and angle

Circles and sectors fight rectangular coordinates, so switch to radius r and angle theta. Substitute x = r cos theta and y = r sin theta, and describe the region with r and theta ranges. Since r squared equals x squared plus y squared, the point (3, 4) has r squared of 25 and r of 5.

**Example.** The key move is the area element: dx dy becomes r dr dtheta, never plain dr dtheta. Mia is right that the extra r appears because polar grid cells get wider as r grows: a cell of fixed angular width has arc side r times angle, so its area grows with r. Forgetting that extra r is the classic error.

The unit disc x squared plus y squared <= 1 becomes simply 0 <= r <= 1 with theta sweeping 0 to 2 pi. Rings, called annuli, use a nonzero inner radius, and slices, called sectors, restrict theta. Whenever the boundary or the integrand involves x squared plus y squared, polar form is worth trying first.

**Tip.** Set up inside out. The inner r integral usually produces powers of r thanks to the extra factor: the integral from 0 to 2 of r dr uses antiderivative r squared over 2, giving 2. The outer theta integral often just multiplies by the swept angle. For z = x squared plus y squared, the integrand becomes r squared, so the volume over the unit disc follows the same pattern.

**Recap.** Substitute r and theta, carry the extra r, set round limits, and integrate inside out.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4)
- [Double Integrals over General Regions](https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw)

## Opens up

- [Change of Variables and the Jacobian](https://lightmysky.com/learn/mathematics/change-of-variables-and-the-jacobian-mt_1_LRE2cJft)
- [Triple Integrals and Coordinates for Solids](https://lightmysky.com/learn/mathematics/triple-integrals-and-coordinates-for-solids-mt_RFeK8LD_jT)