Double Integrals in Polar Coordinates Section 16.4 Double Integrals in Polar Coordinates

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

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

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

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.

EXAMPLE Evaluate where R is the region

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

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.