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
 n  m





f ( x , y , z ) dV  lim      f ( x i , y j , z k )  x   y   z


m


n   k 1  j 1  i 1
p

W
p 
x 
ba
n
, y 
d c
m
' z 
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

f ( x , y , z ) dV 
h
d
 
g
c
b
f ( x , y , z ) dx dy dz
a
W
• 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
1
  
0
1 z
1
1
2
f ( x , y , z ) dy dz dx
0
Example
• Find limits for the integral W f ( x , y , z ) dV
where W is the region shown
z
z
y
x
y
x
This is a quarter sphere of radius 4
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 z = -6 and sides formed by the three
planes y = 0 and y – x = 4 and 2x + y + z =4.
Example
• Calculate the volume of the figure bound by
the following curves
x  y  16
2
2
y 3
z  3 y
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