1. Double Integrals in Polar Coordinates

2. Sometimes equations and regions are expressed more simply in polar rather than rectangular coordinates. Recall: Polar Coordinates:

3. Double Integrals in Polar Coordinates The basic region in polar coordinates is the “Polar Rectangle”: Goal: Compute the double integral where R is a polar rectangle.

4. Double Integrals in Polar Coordinates Therefore a typical Riemann Sum will be

5. Double Integrals in Polar Coordinates Taking the limit as m and n approach infinity, gives the following important result: Note 2: The limits on r may depend on θ: Note 1:

6. Double Integrals in Polar Coordinates- Example 1 Sketch the region of integration and evaluate the integral by changing to polar coordinates. In polar coordinates, the region is bounded by

7. Double Integrals in Polar Coordinates - Example 2 The two surfaces intersect along a curve C: This quarter of a circle of radius 3 is the projection of C on the xy plane and it is also the boundary of the region of integration R. In polar coordinates the region is described by:

8. Double Integrals in Polar Coordinates – Example 3 By symmetry we can compute 8 times the volume in the first octant. Our region of integration is then: The surface is bounded above by

9. Double Integrals over General Regions Example 4 The two surfaces intersect along a curve C. The circle of radius 3 is the boundary of the region of integration R.