ma 242.003

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