 Hello, and welcome to this screencast on section 11.7, triple integrals. In our previous work, we saw that we could determine the mass of a thin object called a lamina using double integrals. This worked because the object is so thin, we assumed the object was contained in two dimensions, and the density was given by the x and y coordinates alone, which is why a double integral applied. If we'd like to determine the mass of a three-dimensional object with varying density, we will also need to consider how the density changes as the z value varies. This suggests that we'll also need to extend our work from using a double integral to a triple integral to account for all three variables, x, y, and z. We proceed by defining both a triple Riemann sum and the corresponding triple integral. Consider a function f of three variables defined on a box, capital B. To construct the triple Riemann sum, we use a very familiar procedure. We subdivide the x interval into m equal size pieces. We also subdivide the y interval into n subintervals. We subdivide the z interval into l subintervals. These subintervals partition the box, capital B, into subboxes, each with volume delta v. We choose a point from each subbox to plug into f and sum up the product of these values with delta v, which gives us the triple Riemann sum for f on b. The triple integral of f over capital B is then defined as the limit of this triple Riemann sum. Now that we've defined the triple integral, let's quickly review some interpretations of the triple integral that extend from our work with double integrals. First, if f is the density function for a three-dimensional solid b, then the triple integral of f gives us the mass of b. The triple integral of the constant one over a solid s gives us the volume of s. The average value of a function f over the solid s is given by the following expression involving a triple integral. And if delta is a density function, the x, y, and z coordinates of the center of mass can be found using the following expressions. One last item to note concerns evaluating triple integrals. In Cartesian coordinates, the volume element dv is dzdy dx. Therefore, the triple integral of a function f over a box can be evaluated as an iterated integral, just like we had for double integrals. If we want to evaluate a triple integral over a solid that is not a box, then we'll need to describe the solid in terms of variable limits. Thank you.