Home»Math Guides»Double Iterated Integrals (polar coordinates for circular region of integration)
Examples of Double Integrals in Polar Coordinates, Equations to Memorize for Polar Coordinates Transformation
We have many worked out examples for solving double integrals using polar coordinates.
In general you just need to set up the problem correctly, make substitutions, get new integral boundaries for radius and theta, then solve it.
You use polar coordinates when the region of integration is circular, or in other words, when you’re confronted with a double integral that has equations that “look like circle” (squared variable terms) in the integral boundaries.
In general, memorize the following formulas for converting to polar coordinates.
\[{r^2} = {x^2} + {y^2}\]
\[x = r\cos (\theta )\]
\[y = r\sin (\theta )\]
\[\frac{y}{x} = \tan (\theta )\]
\[dxdy = rdrd\theta \]
As well, when converting from Cartesian to polar coordinates, you need to multiply the inside of the double integral by an extra “r” term. See the aforementioned link on why you need to do so, it involves a matrix of partial derivatives known as the Jacobian.
You remember how when you used basic substitution you needed to find the derivative? Like, say you wanted to use (u = {x^2}), then you need to also use (du = 2xdx).
Well for polar coordinates or any kind of multi-variable transformation you need to find the Jacobian, which is the determinant of all of the partial derivatives, but when you do so for polar coordinates, you will get \(dxdy = rdrd\theta \), which you can memorize and use for any polar coordinate transformation. You’re generally not expected to prove that each time you do a polar coordinates math problem unless specifically asked to do so.
Explanation on why you need to multiply by “r” when using polar coordinates
Try an example problem where you solve a double integral using polar coordinates!