Examples of Double Integrals in Polar Coordinates

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.

When you're confronted with a double integral that has equations that look like circle, chances are you need to convert it to polar coordinates to solve it. In general, memorize the following formulas for converting to polar coordinates.

As well, when converting from Cartesian to polar coordinates, you need to multiply the inside of the double integral by an extra "r" term. 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 multivariable transformation you need to find the Jacobian, which is the determinant of all of the partial derivatives, but the basic point is that you get \(dxdy = rdrd\theta \), which you can memorize and use for any polar coordinate transformation.

\[{x^2} + {y^2} = {r^2}\]

\[x = r\cos (\theta )\]

\[y = r\sin (\theta )\]

\[\frac{y}{x} = \tan (\theta )\]

\[\theta = {\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\]

\[dxdy = rdrd\theta \]

We have many worked-out examples where double integrals are solved using polar coordinates below:

Double Integrals in Polar Coordinates (Ex1)

Double Integrals in Polar Coordinates (Ex2)

Double Integrals in Polar Coordinates (Ex3)

Double Integrals in Polar Coordinates (Ex4 Cylinders and Ellipsoids)

Double Integrals in Polar Coordinates (Ex5 Spheres and Cylinders)

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