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Triple Integrals in Spherical Coordinates
Recall that the spherical coordinates (
) of a point
in space are shown below, where
is the distance from the origin
to
,
is the same angle as in cylindrical coordinates, and
is the angle between the positive z -axis and the line segment joining the origin and
.
The relationship between rectangular and spherical coordinates is given by the equations :
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and
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In spherical coordinates system the counterpart of a rectangular boxes is a spherical wedge
![]()
as shown below :
It is sometimes difficult to evaluate the triple integral
using Cartesian coordinates. For example,
, where
is the unit ball:
= { (
) |
}
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Note that in the spherical coordinates
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Although we defined the triple integrals by dividing solids into small rectangular boxes, it can be shown that dividing a solid into small spherical wedges always gives the same result.
We can see from the figure above that the length of the navy line segment is
, the length of the green circular segment is
, and the length of the gold circular segment is
, where
.
If
,
, and
are small enough then the spherical wedge is approximately a rectangular box with dimension
,
,
. Hence an approximation of the volume of spherical wedge above is given by
.
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So if we divide
into smaller spherical wedges
by means of equally spaced spheres
, half planes
, and half-cones
, then an approximation of the volume of
is
![]()
and so
is approximately equal to
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Therefore, we have
=
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where
.
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Now let's evaluate the integral
,
where
and
= { (
) |
}.
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=
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=
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=
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Note that it would have bee extremely difficult to evaluate the integral above without spherical coordinates. The iterated integral in rectangular coordinates would have been
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Now consider the solid that lies above the cone
and below the sphere
, as shown below. What is the volume of the solid ?
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Notice that the sphere passes through the origin and has center (
) with radius
. And equation of the sphere in spherical coordinates is given by
or
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The cone can be written as
=
=
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This gives
, or
. Therefore, the description of the solid in spherical coordinates is
![]()
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From the animations above, we see that the volume of the solid is
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