Section 16.3 Triple Integrals

1. Section 16.3Triple Integrals

2. 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

3. If • then • In this case we have a rectangular shaped box region that we are integrating over

4. 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

5. 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)

6. 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

7. 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

8. Example • Calculate the volume of the figure bound by the following curves

9. 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