ma 242.003
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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. - PowerPoint PPT PresentationTRANSCRIPT
MA 242.003
• Day 39 – March 1, 2013• Section 12.4: Double Integrals in Polar Coordinates
Section 12.4Double Integrals in Polar Coordinates
Section 12.4Double Integrals in Polar Coordinates
Motivation: Use double integration to compute the volume of the upper hemisphere of radius 1.
Section 12.4Double Integrals in Polar Coordinates
Motivation: Use double integration to compute the volume of the upper hemisphere of radius 1.
D
Section 12.4Double Integrals in Polar Coordinates
Motivation: Use double integration to compute the volume of the upper hemisphere of radius 1.
D
Section 12.4Double Integrals in Polar Coordinates
Motivation: Use double integration to compute the volume of the upper hemisphere of radius 1.
DFACT: This integral is in fact almost trivial to do in polar coordinates!!
To study polar coordinates to use with double integration we must:
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:
1. Define Polar Coordinates
2. Set up the transformation equations
x
yr
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.
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
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.
3. Study the Polar coordinate Coordinate Curves
The r = constant coordinate curves
The = constant coordinate curves
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
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
And above the x-axis.
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
Area of a polar rectangle
Figure 3. Figure 4
Compute the volume of the upper hemisphere of radius 1
