Section 16 3 triple integrals
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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

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Section 16.3 Triple Integrals

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Section 16 3 triple integrals

Section 16.3Triple 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


Section 16 3 triple integrals

  • If

  • then

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


Section 16 3 triple integrals

  • 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

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)


Example1

Example

  • Sketch the region of integration


Example2

Example

  • Find limits for the integral

    where W is the region shown


Section 16 3 triple integrals

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

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


Example3

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.


Example4

Example

  • Calculate the volume of the figure bound by the following curves


Some notes on triple integrals

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


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