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# Section 16.3 Triple Integrals PowerPoint PPT Presentation

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

Section 16.3 Triple Integrals

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## Section 16.3Triple 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

• then

• 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

• 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

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

### Example

• Find limits for the integral

where W is the region shown

z

z

y

x

y

x

This is a quarter sphere of radius 4

z

z

x

x

y

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

### 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