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