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Triple Integrals in Spherical Coordinates

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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 :

          

and  

In spherical coordinates system the counterpart of a rectangular boxes is a spherical wedge  

as shown below :

3DVIEW

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

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.

3DVIEW

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

=

where .

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Now let's evaluate the integral

,

where   and  = { ( ) | }.

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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

Now consider the solid that lies above the cone  and below the sphere , as shown below. What is the volume of the solid ?

3DVIEW

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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

  =     =  

This gives   , or    . Therefore, the description of the solid in spherical coordinates is

  

    3DVIEW

    3DVIEW

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From the animations above, we see that the volume of the solid is

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