As we saw before the order of integration can be changed. We use an example to show that.
Example
Let E be the region bounded below by the cone z = ⎷x² + y² and above by the sphere z = x² + y² + z².Set up a triple integral in spherical coordinates and find the volume of the region using the following orders of integration.
a. dρdഴdθ
b. dഴd𝜌dθ
Solution
As we have seen previously, in order to change the order of integration of a triple integral of a function over a bounded region, the region has to be projected on one of the three plane coordinate systems of the three-dimensional space coordinates. In this post we learned to project the region E in a different coordinates system when the triple integral becomes difficult to calculate. As the double integral seems to be difficult to calculate, we change from plane to polar coordinates to make the computation easier.
Example
y = x² + z² and the plane y = 4
Solution
In order to define the region E, let's project it on the xy plane. The projection of E onto this plane is the region bounded above by above by the plane y = 4 and below by the parabola y = x² This is shown in the following figure:
According to the projection above, the region E is then determined by:
