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

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

This is a quarter sphere of radius 4 xx xx y y y y z z z z

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