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:
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