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
- It is often easiest to visualize if we put theta as the outside variable.
- 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
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