Section 16.3 Triple Integrals

1. Section 16.3Triple Integrals

2. 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

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 • 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 • Sketch the region of integration

7. Example • Find limits for the integral where W is the region shown

8. z z y x y x This is a quarter sphere of radius 4 z z x x y y

9. 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

10. 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.

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

12. 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