 Hello, and welcome to this screencast on section 11.4, Applications of Double Integrals. In this screencast, we are going to quickly recap some applications of double integrals that are discussed in your textbook. First, we'll see that double integrals can be used to compute the mass of certain objects. If we have a function delta that describes the density of a lamina, which is just a flat, thin object, defined by a planar region D, then the mass is given by the following double integral. We can also use double integrals to calculate area. If we have a closed bounded region D in the plane, the area of D is given by the following double integral. Note that this is actually related to the mass formula from the previous slide. If we assume that such an object has density equal to 1 everywhere, as we have in this integrand, then the object's mass will be equal to the area. Just as we could calculate the mass of a lamina, we can also calculate the coordinates of the center of mass. The coordinates of the center of mass of a lamina D with density function delta are given by the following formulas, which are in terms of double integrals. We can also use double integrals in applications such as probability. Simply put, a joint probability density function is a function that gives us the probability that some random variables take on a specific value. We assume that we have a joint probability density function f, a function of two independent variables x and y, defined on a domain D that satisfies the following conditions. First, f is non-negative for on D. We need this condition because probabilities are always non-negative. Next, the probability that x is between some values a and b and y is between some values c and d is given by the following double integral. Lastly, the probability that a point x, y is in D is equal to 1. This last condition just ensures that all outcomes are possible since the sum of the probabilities of all possible events must equal 1. Such a function as we have here can be used as the joint probability density function for various situations as you will see in our work in section 11.4.