 Hello, and welcome to Screencast 6.4.2. In this screencast, we will consider an example of a physics application and will calculate the amount of work that is done while lifting a chain. In our example, we have a chain that's 45 feet long and weighs 1.5 pounds per foot. The chain is hanging straight down from the top of a building, and our job is to pull all of the chain up to the top. We would like to know how much work we do to lift that chain. If you've studied physics, you may recall that work is equal to force, and in our example that force is measured in pounds, the weight of our chain, multiplied by distance, which we are measuring in feet. At first glance, this may not seem like we need calculus because we're just multiplying two values together. However, as we are lifting the chain, then less chain is hanging over the edge of the building, and the weight of the remaining chain will decrease, so the force in our calculation is not constant. We will create a function, which we're going to call wh, and we'll let that represent the weight of the remaining chain that's hanging over the building after h feet have been pulled up. The output of our function wh will be measured in feet, or sorry, in pounds. When h is equal to zero, and we haven't started pulling up the chain at all, the weight of the chain is our 45 feet multiplied by 1.5 pounds per foot for a total of 67.5 pounds. The wh function is going to be decreasing 1.5 pounds for each foot of chain that we pull up because that means one foot less of chain that's still hanging over the side of the building. So that means our wh function will be a linear function because it has a constant rate of change, and that slope is going to be negative 1.5 pounds per feet because the weight of the chain that's hanging there is going to be decreasing. So we'll expect to have a negative slope. When we put this information together, our result for the weight function is wh is equal to negative 1.5h plus 67.5. So this weight function gives us our force in the calculation above. So next we need to represent the distance that we're lifting the chain. We're going to let delta h be the distance the chain is lifted, just a slice of lift measured in feet. So we're going to imagine just doing it a small distance so that the weight of our chain is constant. So then wh is multiplied by delta h. We multiply these two quantities together. This represents the work done by lifting that h feet of chain just a distance of delta h feet. So again that means we're lifting that h feet of chain just a small distance, just delta h feet. And so we can just multiply that together using our product there. So that is an approximation for just a slice of the entire task of lifting up that chain just delta h feet. So to calculate the exact value of lifting the entire chain we're going to need a definite integral. The length of the chain that we are lifting varies from 0 feet to 45 feet. So the limits on our definite integral are 0 to 45. Our integrand is our weight function negative 1.5h plus 67.5 and our distance delta h becomes dh. Next we need to find an antiderivative for our integrand which results in negative 1.5h squared over 2 using the power rule and we also use that again for the next term and we get 67.5h. Now that we've got our antiderivative we're going to evaluate this at h equals 0 and h equals 45 and then subtract. So when we evaluate our antiderivative at h equals 45 we get negative 0.75 that's our negative 1.5 divided by 2 and we multiply that by 45 squared and then to that we'll add 67.5 times 45. When we evaluated h equals 0 our result is simply 0. Now when we put that first term into our calculator we get negative 1518.75 and then the following term we're adding 3037.5. When we simplify the sum of those two terms our result is positive 1518.75 but now since this is an application we want to remember to put the appropriate units on our answer and so our final result is 1518.75 foot pounds of work that we did in order to lift that entire 45 foot chain up to the top of the building and that's our final result. Thanks for watching.