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Online Notes / Calculus III (Notes) / Multiple Integrals / Triple Integrals in Spherical Coordinates

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In the previous section we looked at doing integrals in terms of cylindrical coordinates and we now need to take a quick look at doing integrals in terms of spherical coordinates.

 

First, we need to recall just how spherical coordinates are defined.  The following sketch shows the relationship between the Cartesian and spherical coordinate systems.

Spherical_G1

 

Here are the conversion formulas for spherical coordinates.

 

 

 

 

We also have the following restrictions on the coordinates.

 

 

 

For our integrals we are going to restrict E down to a spherical wedge.  This will mean that we are going to take ranges for the variables as follows,

 

 

 

 

Here is a quick sketch of a spherical wedge in which the lower limit for both  and  are zero for reference purposes.  Most of the wedges we’ll be working with will fit into this pattern.

 

Spherical_G1

 

From this sketch we can see that E is really nothing more than the intersection of a sphere and a cone.

 

In the next section we will show that

 

 

Therefore the integral will become,

 

 

This looks bad, but given that the limits are all constants the integrals here tend to not be too bad.

 

Example 1  Evaluate  where E is the upper half of the sphere .

Solution

Since we are taking the upper half of the sphere the limits for the variables are,

                                                                 

 

The integral is then,

                                

 

Example 2  Convert  into spherical coordinates.

Solution

Let’s first write down the limits for the variables.

                                                  

 

The range for x tells us that we have a portion of the right half of a disk of radius 3 centered at the origin.  Since we are restricting y’s to positive values it looks like we will have the quarter disk in the first quadrant.  Therefore since D is in the first quadrant the region, E, must be in the first octant and this in turn tells us that we have the following range for  (since this is the angle around the z-axis).

                                                                  

Now, let’s see what the range for z tells us.  The lower bound, , is the upper half of a cone.  At this point we don’t need this quite yet, but we will later.  The upper bound, , is the upper half of the sphere,

                                                            

and so from this we now have the following range for  

                                                          

 

Now all that we need is the range for .  There are two ways to get this.  One is from where the cone and the sphere intersect.  Plugging in the equation for the cone into the sphere gives,

                                                        

 

Note that we can assume z is positive here since we know that we have the upper half of the cone and/or sphere.  Finally, plug this into the conversion for z and take advantage of the fact that we know that  since we are intersecting on the sphere.  This gives,

                            

 

So, it looks like we have the following range,

                                                                 

 

The other way to get this range is from the cone by itself.  By first converting the equation into cylindrical coordinates and then into spherical coordinates we get the following,

                                 

 

So, recalling that , the integral is then,

                


Online Notes / Calculus III (Notes) / Multiple Integrals / Triple Integrals in Spherical Coordinates

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