 Hi everyone and welcome to this lesson on volumes of solids of revolution, the washer method. This is the counterpart to the disc method. Think of a disc but with a hole in it. So when we say washer, we mean like the hardware kind of washer. Or if you prefer food analogies, think of a lifesaver or a donut. That's the type of washer we're talking about. So let me take you to a couple of applets first to give you more of a visualization to set the stage for what it is we're talking about. So this first one is from mathdemos.org. If you take a look at the animation there, you can see that the washer is forming. It definitely looks like a disc but it has that big hole in the middle of it. So what we're going to have to do is come up with a formula to account for the hole being there because we don't want to include the area of the hole when we're finding our formula that we're going to use. Once again we're going to be using the idea that was built upon Riemann sums and the using the rectangles underneath a curve. And here once again we're going to have representative rectangles that we will think of rotating that representative rectangle about an axis of revolution and that's what's going to create our washer. If you go to this website I purposely left the URL there for you at the top in case you wanted to investigate it on your own. If you scroll down they have a lot more other pictures and demonstrations of the types of things we're talking about. There's our hardware washers right there. There's your good old lifesaver and the donut. I guess you could use a bagel too right? So they give a lot of different visualizations you can even think of like a DVD or a CD type of disc that has the hole in the middle there. So this is a great website if you care to explore it later on. Let me take you to another visualization. This is from calculusapplets.com and if you take a look at it you can see the rough outline of what looks to be a cylinder but there is a hole going down the cylinder there. So imagine taking this yellow region and rotating it around this black x-axis right here going through the middle. Alright so what you can see here is the representative rectangle that we'll be talking about in our lesson today that dark gray part and imagine flipping that around the x-axis and that's going to create the gray washer that you see here. Alright so that's what we're going to have to come up with a formula for as that's going to become our integrand for our definite integral. So let's go back to our lesson. So if we think once again of how in general we get volume of anything remember it's going to be the area of the base times the height the thickness. So just like we did before our volume calculations are going to be doing a definite integral of the area of in this case the washer face times the thickness dx if it's in terms of a different orientation of our representative rectangle that integrand is going to be in terms of y. So very similar to what we saw with disks. As a reminder again a solid of revolution is a solid generated by revolving a region about a line which we refer to as the axis of revolution. The solids we're typically going to be dealing with will involve a region bounded by a curve the x-axis and or the y-axis and perhaps some horizontal or vertical line. What we need to do is find the volume of the solid disk and take away from that the volume of the hole. So let's try to replicate that gray washer we saw on that last applet from calculusapplets.com. So here's the outer edge of our washer and here's the hole in the middle. Basically what we want is just this part. So we need to figure out how to do that and again we're going to build upon the fact that volume of anything is generally the area of the base so we need that area of that shaded region I just shaded in for you times the thickness all right well it's a circle so we know pi r squared. So if we thought of big r as being the radius to the outside circle so that's big r and little r as the radius of the hole we could come up with an expression for the area of the entire outside circle and then subtract from that the area of the inside circle so that would look like pi big r squared minus pi little r squared now that's just the area of the face of the surface remember to get volume we need to multiply this by the height the thickness which we're typically going to refer to as either dx or dy and we're going to turn it into a definite integral all right so typically what we're going to use notice you can factor out a pi here so the basic formula we're generally going to use is pi times the quantity big r squared minus little r squared minus little r squared and then either dx or dy depending upon the orientation of our representative rectangle so let's talk first about solids that are revolved about a horizontal axis when we revolve about a horizontal axis I have a picture for you on the next slide it looks like this we end up with a vertical representative rectangle so your representative rectangle would maybe go something like this all right so that would be a dx because the orientation of our representative rectangle is vertical that turns it into a dx problem so let me scoot back for a second the outer radius capital r is the distance from the axis of revolution to the far side of the curve of the rectangle which you'll see here capital r so it's going from the axis of revolution which in this case is the dotted line there to the far side of your rectangle the inner radius r is the distance from the axis of revolution to the close side of the curve or the close side of your representative rectangle all right so you'll notice little r here going from the dotted line again your axis of revolution to this closest side of the representative rectangle then you can see how you set up your volume integral again axis your limits of integration need to be x values because it's a dx problem now if we have a vertical axis of revolution that's going to turn it into a horizontal representative rectangle so it would be going something like this and that turns it into a dy problem so capital r is once again the outer radius the distance from the axis of revolution to the far side of the curve so once again from the dotted line which is your axis of revolution to the far side of your representative rectangle little r is the inner radius that is the distance from the axis of revolution to the close side of the curve or representative rectangle again little r is really the radius of the hole itself so you can see how the volume integral gets set up if it's a dy problem your limits of integration need to be y values but still you have that pi times big r squared minus little r squared you do just need to remember that your integrand here needs to be in terms of y now I want to point out to you two major characteristics of a washer method problem and think of this picture and this one notice in both cases your representative rectangle is perpendicular to the axis of revolution but it is not touching it that is contrast to the disc method which the representative rectangles were still perpendicular to your axis of revolution but with a disc method those rectangles were touching the axis of revolution in a washer method they are not so that is really those two characteristics being perpendicular to your axis of revolution and then whether it touches the axis or not that's really what differentiates a disc method problem from being a washer method problem as you go about finding your capital r and lower case r you're still going to adhere to that top minus bottom and right minus left ideas so as you work through practice problems keep that in mind that as you find your your expressions for capital r and little r you're always doing top minus bottom or right minus left