Finding a volume with triple integrals in two ways
Let
E be the region bounded below by the
-plane, above by the sphere
and on the sides by the cylinder
(
[link] ). 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:
Note that the equation for the sphere is
and the equation for the cylinder is
Thus, we have for the region
Hence the integral for the volume is
Since the sphere is
which is
and the cylinder is
which is
we have
that is,
Thus we have two regions, since the sphere and the cylinder intersect at
in the
-plane
In three-dimensional space
in the spherical coordinate system, we specify a point
by its distance
from the origin, the polar angle
from the positive
(same as in the cylindrical coordinate system), and the angle
from the positive
and the line
(
[link] ). Note that
and
(Refer to
Cylindrical and Spherical Coordinates for a review.) Spherical coordinates are useful for triple integrals over regions that are symmetric with respect to the origin.
Recall the relationships that connect rectangular coordinates with spherical coordinates.
From spherical coordinates to rectangular coordinates:
From rectangular coordinates to spherical coordinates:
Other relationships that are important to know for conversions are
and
The following figure shows a few solid regions that are convenient to express in spherical coordinates.
Integration in spherical coordinates
We now establish a triple integral in the spherical coordinate system, as we did before in the cylindrical coordinate system. Let the function
be continuous in a bounded spherical box,
We then divide each interval into
subdivisions such that
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