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Triple Integrals in
Spherical Coordinates

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
...
The following sketch shows
the relationship between the Cartesian and spherical coordinate systems
...


x = ρ sin ϕ cos θ

y = ρ sin ϕ sin θ

z = ρ cos ϕ

x2 + y2 + z 2 = ρ 2
We also have the following restrictions on the coordinates
...
This will mean that we
are going to take ranges for the variables as follows,

a≤ ρ ≤b

α ≤θ ≤ β
δ ≤ϕ ≤γ
Here is a quick sketch of a spherical wedge in which the lower limit for both ρ and ϕ are zero
for reference purposes
...


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


Example 1 Evaluate

2
2
2
16
z
dV
where
E
is
the
upper
half
of
the
sphere
x
+
y
+
z
= 1
...


Solution
Let’s first write down the limits for the variables
...
Since we are restricting y’s to positive values it looks like we will have the quarter disk in
the first quadrant
...


0 ≤θ ≤

π

2

Now, let’s see what the range for z tells us
...
At this point we don’t need this quite yet, but we will later
...
There are two ways to get this
...
Plugging in the equation for the cone into the sphere gives,

(

x +y
2

2

)

2

+ z 2 = 18

z 2 + z 2 = 18
z2 = 9
z =3
Note that we can assume z is positive here since we know that we have the upper half of the cone
and/or sphere
...
This gives,

ρ cos ϕ = 3

3 2 cos ϕ = 3
1
2
cos ϕ =
=
2
2

ϕ=

⇒

So, it looks like we have the following range,

0≤ϕ ≤

π
4

π
4

The other way to get this range is from the cone by itself

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