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We integrate over regions in spherical coordinates.
Another way to generalize polar coordinates to three dimensions is with spherical
An ordered triple consisting of a radius, an angle, and a height can be graphed as
Coordinates of this type are called spherical coordinates.
Consider the point in spherical coordinates. What is this point when expressed in
Triple integrals in spherical coordinates
If you want to evaluate this integral you have to change to a region defined in
-coordinates, and change to some combination of leaving you with some iterated
integral: Now consider representing a region in spherical coordinates and
let’s express in terms of , , and . To do this, consider the diagram below:
Here we see
Recalling that the determinate of a matrix gives the volume of a
parallelepiped, we could also deduce the correct form for by setting
We may now state at theorem:
Fubini Let be continuous on the region Then:
Write down a triple integral in spherical coordinates that will compute the volume of
a sphere of radius .