Goal: Evaluate a double integral over a polar rectangular region

Definition

The double integral of a function f(r, θ) over a polar rectangular plane in the θ plane is defined as 

As in double integrals over a rectangular region, the double integral over a polar rectangular region can be expressed as in iterated integral in polar coordinates.

In polar coordinates dA is replaced by rdrdθ. The double integral of f(x,y) in rectangular coordinates can be expressed in polar coordinates by substitution. This is done by substituting x by rcosθ, y by rsinθ and dA by rdrdθ.

The properties for double integrals over rectangular coordinates apply also to the double integrals over polar coordinates.

Example 1

Solution

As we can see in the figure below, r =1 and r = 3 represent circles of radius r =1 and r =3 and 0≤ θ ≤ℼ covers the entire top half of the plane.

Example 2

Solution

The figure is similar to that in example 1 but with outer radius 2. Do it by yourself.

Practice