2009-11-04: Triple Integrals in Spherical Coordinates and Changing Variables

Stewart section 16.8--16.9

• Last time we introduced spherical coordinates. An example: a spherically bent brick.
• Game: Describe the region contained by the brick in spherical coordinates.
• Note that if the density d(x,y,z) = xy + z, it is not constant in any of the coordinates, and so we need a triple integral to find the mass of the brick.
• The mass of a small piece of the brick is d(x,y,z) dV, so the total mass is...
• We can visualize this sum (that is, the integral) by looking at a slice with theta fixed.
• Key Point
1. It is often easiest to visualize if we put theta as the outside variable.
2. If either cylindrical or spherical coordinates are appropriate, it's usually better to use cylindrical coordinates.
• Finally, a word about section 16.9. Why is the polar dA = r dr dtheta? Answer: because it's the area of a parallelogram.
ma215-080-f09 lecture outline 2009-11-04
Created: Wed Nov 4 13:25:26 EST 2009