Section 16.3Triple Integrals

• A continuous function of 3 variable 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 volume in each of these cubes

p

k

n

j

m

ikji

pnm

W

zyxzyxfdVzyxf1 1 1

),,(lim),,(

•If

then

•In this case we have a rectangular shaped box region that we are integrating over

p

ghz

m

cdy

n

abx

',

},,),,{( hzgdycbxazyxW

• We can compute this with an iterated integral– In this case we will have a triple integral

• Notice that we have 6 orders of integration possible for the above iterated integral

• Let’s take a look at some examples

h

g

d

c

b

aW

dzdydxzyxfdVzyxf ),,(),,(

Example• Pg. 801, #3 from the text, Find the triple integral

W is the rectangular box with corners at (0,0,0), (a,0,0), (0,b,0), and (0,0,c)

zyxezyxf ),,(

Example

• Pg. 801, #5 from the text, Sketch the region of integration

• Let’s set up the limits of integration for #15 on pg 801

21 1 1

0 1 0( , , )

zf x y z dydzdx

Triple Integrals can be used to calculate volume

• Pg. 801, #18 from the text

• 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• Calculate the volume of the figure bound by

the following curves

yz

yz

y

yx

23

3

3

1622

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 integrals