1. Double Integrals in Polar Coordinates
Section 16.4
Double Integrals in
Polar Coordinates

2. POLAR RECTANGLES
In polar coordinates, a polar rectangle R has the form
R = {(r, θ) : a ≤ r ≤ b, α ≤ θ ≤ β}

3. CONVERTING TO POLAR COORDINATES
Partition R into small polar rectangles given by Rij = {(r, θ) | ri − 1 ≤ r ≤ ri, θj − 1 ≤ θ ≤ θj} The area of rectangle Rij is given by
where is the average radius of the rectangle.
Then the typical Riemann sum is

4. CONVERTING TO POLAR COORDINATES
If we write g(r, θ) = r f (r cos θ, r sin θ), then the Riemann sum can be written as
which is a Riemann sum for the double integral

5. CHANGE TO POLAR COORDINATES IN A DOUBLE INTEGRAL
If f is continuous on the polar rectangle R given by 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β, where 0 ≤ β − α ≤ 2π, then
NOTE: Be careful not to forget the additional factor r on the right side of the formula.

6. EXAMPLE
Evaluate where R is the region

7. D = {(r, θ) | α ≤ θ ≤ β, h1(θ) ≤ r ≤ h2(θ)}.
AN EXTENSION
If f is continuous on a polar region of the form
D = {(r, θ) | α ≤ θ ≤ β, h1(θ) ≤ r ≤ h2(θ)}.
then

8. EXAMPLES
1. Use a double integral to find the area enclosed by one leaf of the three-leaved rose r = 3 sin 3θ.
2. Compute where D is the region in
the first quadrant that is outside the circle r = 2 and inside the cardiod r = 2 + 2cos θ.
3. Compute the volume of the solid that lies under
the hemisphere , above the xy-plane, and inside the cylinder x2 + y2 − 4x = 0.