Section 15.4
Double Integrals in Polar Coordinates
When the region is round, use polar coordinates.
1
Introduction
If we try to integrate over a circle using rectangular coordinates, we get nasty limits like .
Polar coordinates () turn circles into "rectangles" in the -plane, simplifying the limits to constants.
2
The Polar Formula
Change of Variables
Use and .
Comparison: Don't forget the extra ! .
3
Visualizing Polar Area
Interactive: Paraboloid Volume
4
Worked Examples
Example 1: Volume of Paraboloid
Find the volume under and above the xy-plane.
1. Setup Boundaries:
Intersection with z=0 implies (radius 3).
, .
2. Convert Function:
.
3. Integrate:
.
Example 2: Region between Circles
Evaluate where R is between circles of radius 1 and 2 in 1st quadrant.
1. Bounds:
1st Quadrant: .
Radii: .
2. Integral:
.
3. Solve:
For inner: .
.
Outer: .
Example 3: The Gaussian Integral
Prove that .
1. Square it:
Let . Compute .
.
2. Switch to Polar:
implies .
.
3. Solve:
.
.
Result: .
5
Practice Quiz
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