 Hello, and welcome to a screencast today about finding the work done pumping fluid. We have an upright cylindrical tank with a height of 10 feet and a diameter of 8 feet, and it contains water with a density of 6.24 pounds per feet cubed. If the water in the tank is 6 feet deep, how much work is done to pump the water out of the tank? Okay, so let's go ahead and draw ourselves a tank, and we're going to go ahead and draw in our cylinder. Okay, so we'll pretend like that's a nice-looking cylinder, and we know that the height of this cylinder is 10 feet, and we know that the diameter of the cylinder is 8 feet. Okay, so let me go ahead and draw in my slice of water because we know that the water here is 6 feet deep. Okay, so my slice of my water is going to look something like that, and just like we've done, you know, many times before these problems is we're going to think about what's going to happen with this one slice, and then we can think about what's going to happen when we add up all those slices. Okay, so again we're going to be gunning for an integral here. Let's call the width of my slice delta x just to give it something to call it, and then we know that the radius of the slice is going to be 4. Okay, now we know the radius is a constant only because as you go from the bottom of the cylinder to the top of the cylinder, that radius doesn't change, right? It stays exactly the same. That's not always going to be the case, so if this were a cone or if this were any kind of a shape that had a slant to the side, this radius is going to have to be a function. Okay, so just fair warning there. Okay, so let's figure out what the volume of one of our slices is. Well, that's going to look like a cylinder, so that's going to be pi r squared h, and then as we already talked about, pi r radius is 4 squared, and then my h is then going to be this delta x. Okay, so pretty that up a little bit. That's going to be 16 pi delta x, and then the units on this, so my radius is going to be measured in feet, right? Because my diameter is measured in feet, so I'm going to square that, and then my delta x is also going to be measured in feet because that's a little bit of that height that's measured in feet. So anyway, all those feet together is going to end up giving me feet cubed. Okay, now what's the force on one of these slices? Because that water has a weight to it. Okay, so our density, whatever way you want to look at it. So, rho times v, just like we did in a previous screencast here, so that's going to end up giving us 62.4 pounds per feet cubed times my volume that I just found, which is 16 pi delta x feet cubed. Okay, so again, those feet cubes are going to cancel. That works out fairly nicely. So in here, it just depends on how nice you want to keep these answers. Pi is always an approximation, so I'm going to leave the pi in my answer. So when I multiply the 62.4 and my 16 together, since that's all I really have left, I have 998.4 pi delta x, and then that's going to be measured in pounds. All right, I've got the volume of my slice, which then led me to the force on my slice. So now I have to figure out what's the distance one of these slices is going to travel. Okay, you have a couple of options on how you want to think about this. So the way that I usually look at it is, if my slice is kind of here in the random area, how far does my slice have to travel to get out of the top of the tank? Okay, so if this were at 5, it would have to travel 5 feet. If I were at 3 feet, it would have to travel 7 feet. So the distance that you're traveling of the slice is going to be 10 minus wherever you're at. So that's going to be x. Okay, fantastic. So now we can figure out the work done on a slice. So work is going to be this force on the slice times the distance of our slice. That's why we took the time to write all these things out. So then we're going to end up doing, oops, 998.4 pi delta x, and we're going to be multiplying that by 10 minus x. In this distance, since obviously we're traveling in feet, it's also going to be in feet just so we know. Okay, so our work is eventually going to be a foot. Okay, now this is just the work done on one little slice. So what happens when I add up all those little slices? Alrighty, so I'm going to add up an integral since we're going to be adding up a bunch of different pieces together. I'm going to be integrating all of this stuff here that we just found for our work. So I'm going to go ahead and factor that 998.4 pi out front since it's a constant, and it's just going to make our integral look a lot nicer. Okay, then I'm going to have 10 minus x, and then we're going to be integrating with respect to x. So instead of writing delta x as I almost started to do there, let's go ahead and change that into a dx since we're going to be integrating. Okay, now what are the limits on our integral? So my water is going from 0 to 6 feet, right? So that means that that's exactly where I'm going to be integrating. So at the bottom of this tank is 0 feet. At the top of my water level is going to be 6 feet. So that's going to be the endpoints on my integral. Now, had you done your distance formula up here a little bit differently, the endpoints on your integral are going to be different. Okay, so you want to make sure that you're consistent about those types of things. All right, so this is a fairly straightforward integral to do. So we have our 998.4 pi out front, then we end up doing fundamental theorem of calculus, and we get an anti derivative of 10x minus a half x squared, and then we're going to integrate or we're going to evaluate that from 0 to 6. So plugging all that stuff in, we end up getting 998.4 pi times 42. And again, just because I like to leave my answers as exact as possible, that's going to give me then a 41,932.8 pi feet pounds. Okay, oops, sorry about that. So kind of to recap these problems, honestly doing these three steps right here is so important. Okay, I know it takes time to do, but figuring out the volume of your slice, figuring out the force on your slice, and figuring out the distance of your slice is going to make your integral so much easier to set up and do. So don't let these problems overwhelm you, just take them piece by piece, and hopefully you'll be very successful when you do them. Thank you for watching.