Lecture 18: Triple Integrals, Cyclindrical Coordinates, and

Spherical Coordinates

Part I: Triple Integrals

Objectives

• Objectives: Know how to compute and use triple integrals

Corresponding Section in Simmons: 20.5

Review of Double Integrals• Recall: Double integrals measure volume• is computed by doing the integrals one at a

time. This corresponds to finding the volume in each slice from to , which is , and then adding these sums up

Triple Integrals• Triple integrals measure 4-dimensional volume.• is computed by doing the integrals one at a

time. This corresponds to finding the volume in each slice from to , which is , and then adding these sums up

Example:• What is the 4-dimensional volume of the solid

defined by the equations ?• The 4-dimensional volume of each piece is

Part II: Cylindrical and Spherical Coordinates

Objectives

• Objectives: Know what cylindrical and shperical coordinates are and how to convert between these coordinate systems and Cartesian coordinates

Corresponding Section in Simmons: 20.6

Cylindrical Coorindates• Same as polar coordinates, just with a z-

coordinate as well

Spherical Coordinates

Picture for Spherical Coordinates

• (is the angle on the sphere from the z-axis)• Note: is similar to latitude except that it is at

the North pole and degrees at the equator rather than the other way around.

• Think of as longitude (the horizontal angle on the sphere)