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MA 242.003

MA 242.003 . Day 39 – March 1, 2013 Section 12.4: Double Integrals in Polar Coordinates. Section 12.4 Double Integrals in Polar Coordinate s. Section 12.4 Double Integrals in Polar Coordinates. Motivation : Use double integration to compute the volume of the upper hemisphere of radius 1.

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MA 242.003

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  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 Coordinates

  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. FACT: This integral is in fact almost trivial to do in polar coordinates!! D

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

  8. To study polar coordinates to use with double integration we must: 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 r y x

  11. 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.

  12. 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

  13. 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.

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

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

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

  17. And above the x-axis.

  18. 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

  19. Area of a polar rectangle

  20. Figure 3. Figure 4

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

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