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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

- 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)

- Sketch the region of integration

- 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

- 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

- 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.

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

- 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