1. MA 242.003 Day 39 – March 1, 2013 Section 12.4: Double Integrals in Polar Coordinates

2. Section 12.4 Double Integrals in Polar Coordinate s

3. Section 12.4 Double Integrals in Polar Coordinates Motivation: Use double integration to compute the volume of the upper hemisphere of radius 1.

4. Section 12.4 Double Integrals in Polar Coordinates Motivation: Use double integration to compute the volume of the upper hemisphere of radius 1. D

5. Section 12.4 Double Integrals in Polar Coordinates Motivation: Use double integration to compute the volume of the upper hemisphere of radius 1. D

6. Section 12.4 Double Integrals in Polar Coordinates Motivation: Use double integration to compute the volume of the upper hemisphere of radius 1. D FACT: This integral is in fact almost trivial to do in polar coordinates!!

7. To study polar coordinates to use with double integration we must:

8. 1. Define Polar Coordinates 2. Set up the transformation equations 3. Study the Polar coordinate Coordinate Curves 4. Define the area element in Polar Coords:

9. 1. Define Polar Coordinates

10. 2. Set up the transformation equations x y r

15. 3. Study the Polar coordinate Coordinate Curves Definition: A coordinate curve (in any coordinate system) is a curve traced out by setting all but one coordinate constant, and then letting that coordinate range over its possible values.

16. 3. Study the Polar coordinate Coordinate Curves Definition: A coordinate curve (in any coordinate system) is a curve traced out by setting all but one coordinate constant, and then letting that coordinate range over its possible values. Example: The x = 1 coordinate curve in the plane

17. 3. Study the Polar coordinate Coordinate Curves Definition: A coordinate curve (in any coordinate system) is a curve traced out by setting all but one coordinate constant, and then letting that coordinate range over its possible values. Example: The x = 1 coordinate curve in the plane Definition: A rectangle is a region enclosed by two pairs of congruent coordinate curves.

18. 3. Study the Polar coordinate Coordinate Curves The r = constant coordinate curves The = constant coordinate curves

19. 3. Study the Polar coordinate Coordinate Curves The r = constant coordinate curves The = constant coordinate curves Definition: A rectangle is a region enclosed by two pairs of congruent coordinate curves. Circles Rays

20. 3. Study the Polar coordinate Coordinate Curves The r = constant coordinate curves The = constant coordinate curves Definition: A rectangle is a region enclosed by two pairs of congruent coordinate curves. Circles Rays A Polar Rectangle

23. And above the x-axis.

26. 4. Define the area element in Polar Coords: We use the fact that the area of a sector of a circle of radius R with central angle is

27. Area of a polar rectangle

28. Figure 3.Figure 4

38. Compute the volume of the upper hemisphere of radius 1