Transcript Section 16.3 Triple Integrals
Section 16.3
Triple Integrals
• A continuous function of 3 variables can be integrated over a solid region,
W
, in 3-space just as a function of two variables can be integrated over a flat region in 2-space • We can create a Riemann sum for the region
W
– This involves breaking up the 3D space into small cubes – Then summing up the functions value in each of these cubes
•If
W
{(
x
,
y
,
z
)
a
x
b
,
c
y
d
,
g
z
h
} then
W f
(
x
,
y
,
z
)
dV
lim
m n p
k
1
j n
1
i m
1
f
(
x i
,
y
j
,
z k
)
x
y
z
x
b
a
,
n
y
d
c
'
z m
h
g p
•In this case we have a rectangular shaped box region that we are integrating over
• We can compute this with an iterated integral – In this case we will have a
triple integral
W f
(
x
,
y
,
z
)
dV
d h
g c a b f
(
x
,
y
,
z
)
dx dy dz
• Notice that we have 6 orders of integration possible for the above iterated integral • Let’s take a look at some examples
Example
• Find the triple integral
f
(
x
,
y
,
z
)
e
x
y
z W
is the rectangular box with corners at (0,0,0), (
a
,0,0), (0,
b
,0), and (0,0,
c
)
Example
• Sketch the region of integration 0 1 1
1 0 1
z
2
Example
• Find limits for the integral
W
where W is the region shown
z z x y x
This is a quarter sphere of radius 4
y z z x y x y
Triple Integrals can be used to calculate volume
• Find the volume of the region bounded by
z
=
x
+
y
,
z
= 10, and the planes
x
= 0,
y
= 0 • Similar to how we can use double integrals to calculate the area of a region, we can use triple integrals to calculate volume – We will set
f
(
x
,
y
,
z
) = 1
Example
• Find the volume of the pyramid with base in the plane planes
y z
= -6 and sides formed by the three = 0 and
y
–
x
= 4 and 2
x
+
y
+
z
=4.
Example
• Calculate the volume of the figure bound by the following curves
x
2
y
3
y
2
z
3
y
16
z
3 2
y
Some notes on triple integrals
• Since triple integrals can be used to calculate volume, they can be used to calculate total mass (recall Mass = Volume * density) and center of mass • When setting up a triple integral, note that – The outside integral limits must be constants – The middle integral limits can involve only one variable – The inside integral limits can involve two variables