MA 242.003

• Day 45 – March 18, 2013• Section 9.7: Cylindrical Coordinates• Section 12.8: Triple Integrals in Cylindrical Coordinates

Section 12.8Triple Integrals in Cylindrical Coordinates

Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry.

Section 12.8Triple Integrals in Cylindrical Coordinates

Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry.

Cylinders

Section 12.8Triple Integrals in Cylindrical Coordinates

Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry.

CylindersCones

To study cylindrical coordinates to use with triple integration we must:

1. Define Cylindrical Coordinates (section 9.7)

1. Define Cylindrical Coordinates (section 9.7)

2. Set up the transformation equations

To study cylindrical coordinates to use with triple integration we must:

1. Define Cylindrical Coordinates (section 9.7)

2. Set up the transformation equations

3. Study the cylindrical coordinate Coordinate Surfaces

To study cylindrical coordinates to use with triple integration we must:

1. Define Cylindrical Coordinates (section 9.7)

2. Set up the transformation equations

3. Study the cylindrical coordinate Coordinate Surfaces

4. Define the volume element in cylindrical coordinates:

To study cylindrical coordinates to use with triple integration we must:

1. Define Cylindrical Coordinates (section 9.7)

2. Set up the transformation equations

3. Study the cylindrical coordinate Coordinate Surfaces

4. Define the volume element in cylindrical coordinates:

recall the polar coordinate area element:

1. Define Cylindrical Coordinates

2. Set up the Transformation Equationsa. To transform integrands to cylindrical coordinatesb. To transform equations of boundary surfaces

2. Set up the Transformation Equationsa. To transform integrands to cylindrical coordnatesb. To transform equations of boundary surfaces

2. Set up the Transformation Equationsa. To transform integrands to cylindrical coordinatesb. To transform equations of boundary surfaces

3. Study the Cylindrical coordinate Coordinate Surfaces

Definition: A coordinate surface (in any coordinate system) is a surface traced out by one coordinate constant, and then letting the other coordinates range over their possible values.

Example: The x = 1 coordinate surface is a plane

3. Study the cylindrical coordinate Coordinate Surfaces

Example: The x = 1 coordinate surface is a plane

Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.

Definition: A coordinate surface (in any coordinate system) is a surface traced out by one coordinate constant, and then letting the other coordinates range over their possible values.

3. Cylindrical coordinate Coordinate Surfaces

The r = constant coordinate surfaces

The = constant coordinate surfaces

The z = constant coordinate surfaces

3. Cylindrical coordinate Coordinate Surfaces

The = constant coordinate surfaces

3. Cylindrical coordinate Coordinate Surfaces

Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.

3. Cylindrical coordinate Coordinate Surfaces

Definition: A rectangular box is a region enclosed by three pairs of congruent coordinate surfaces.

A rectangular box in Cartesian coordinates

3. Cylindrical coordinate Coordinate Surfaces

Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.

A rectangular box in Cartesian coordinates

A cylindrical box in cylindrical coordinates

4. Define the volume element in cylindrical coordinates:

Section 12.8Triple Integrals in Cylindrical Coordinates

Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry.

CylindersCones

z

z

(Continuation of example)

(Continuation of example)

(Continuation of example)

(Continuation of example)