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Washer method rotating around non-axis

Washer method when rotating around a horizontal line that is not the x-axis. Created by Sal Khan.
Video transcript
Now let's do a really interesting problem. So I have y equals x, and y is equal to x squared minus 2x right over here. And we're going to rotate the region in between these two functions. So that's this region right over here. And we're not going to rotate it just around the x-axis, we're going to rotate it around the horizontal line y equals 4. So we're going to rotate it around this. And if we do that, we'll get a shape that looks like this. I drew it ahead of time, just so I could draw it nicely. And as you can see, it looks like some type of a vase with a hole at the bottom. And so what we're going to do is attempt to do this using, I guess you'd call it the washer method which is a variant of the disk method. So let's construct a washer. So let's look at a given x. So let's say an x right over here. So let's say that we're at an x right over there. And what we're going to do is we're going to rotate this region. We're going to give it some depth, dx. So that is dx. We're going to rotate this around the line y is equal to 4. So if you were to visualize it over here, you have some depth. And when you rotate it around, the inner radius is going to look like the inner radius of our washer. It's going to look something like that. And then the outer radius of our washer is going to contour around x squared minus 2x. So it's going to look something-- my best attempt to draw it-- it's going to look something like that. And of course, our washer is going to have some depth. So let me draw the depth. So it's going to have some depth, dx. So this is my best attempt at drawing some of that the depth. So this is the depth of our washer. And then just to make the face of the washer a little bit clearer, let me do it in this green color. So the face of the washer is going to be all of this business. All of this business is going to be the face of our washer. So if we can figure out the volume of one of these washers for a given x, then we just have to sum up all of the washers for all of the x's in our interval. So let's see if we can set up the integral, and maybe in the next video we'll just forge ahead and actually evaluate the integral. So let's think about the volume of the washer. To think about the volume of the washer, we really just have to think about the area of the face of the washer. So area of "face"-- put face in quotes-- is going to be equal to what? Well, it would be the area of the washer-- if it wasn't a washer, if it was just a coin-- and then subtract out the area of the part that you're cutting out. So the area of the washer if it didn't have a hole in the middle would just be pi times the outer radius squared. It would be pi times this radius squared, that we could call the outer radius. And since it's a washer, we need to subtract out the area of this inner circle. So minus pi times inner radius squared. So we really just have to figure out what the outer and inner radius, or radii I should say, are. So let's think about it. So our outer radius is going to be equal to what? Well, we can visualize it over here. This is our outer radius, which is also going to be equal to that right over there. So that's the distance between y equals 4 and the function that's defining our outside. So this is essentially, this height right over here, is going to be equal to 4 minus x squared minus 2x. I'm just finding the distance or the height between these two functions. So the outer radius is going to be 4 minus this, minus x squared minus 2x, which is just 4 minus x squared plus 2x. Now, what is the inner radius? What is that going to be? Well, that's just going to be this distance between y equals 4 and y equals x. So it's just going to be 4 minus x. So if we wanted to find the area of the face of one of these washers for a given x, it's going to be-- and we can factor out this pi-- it's going to be pi times the outer radius squared, which is all of this business squared. So it's going to be 4 minus x squared plus 2x squared minus pi times the inner radius-- although we factored out the pi-- so minus the inner radius squared. So minus 4 minus x squared. So this will give us the area of the surface or the face of one of these washers. If we want the volume of one of those watchers, we then just have to multiply times the depth, dx. And then if we want to actually find the volume of this entire figure, then we just have to sum up all of these washers for each of our x's. So let's do that. So we're going to sum up the washers for each of our x's and take the limit as they approach zero, but we have to make sure we got our interval right. So what are these-- we care about the entire region between the points where they intersect. So let's make sure we get our interval. So to figure out our interval, we just say when does y equal x intersect y equal x squared minus 2x? Let me do this in a different color. We just have to think about when does x equal x squared minus 2x. When are our two functions equal to each other? Which is equivalent to-- if we just subtract x from both sides, we get when does x squared minus 3x equal 0. We can factor out an x on the right hand side. So this is going to be when does x times x minus 3 equal zero. Well, if the product is equal to 0, at least one of these need to be equal to 0. So x could be equal to 0, or x minus 3 is equal to 0. So x is equal to 0 or x is equal to 3. So this is x is 0, and this right over here is x is equal to 3. So that gives us our interval. We're going to go from x equals 0 to x equals 3 to get our volume. In the next video, we'll actually evaluate this integral.