In a previous example I show how to set up a triple integral in three ways. Calculating a volume in three ways comes to using the same procedure. The following example shows how to calculate the volume with triple integrals in three ways.
Example
Let E be the region bounded below by the rθ plane, above by the sphere x² + y² + z² = 4 and on the sides by the cylinder x² + y² = .1. Set up a triple integral in cylindrical coordinates to find the volume of the region using the following orders of integration, and in each case find the volume and check that the answers are the same.
a. dzdrdθ
b.drdzdθ
Solution
Practice
Redo the previous example with the following order dθdzdr,
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