 Hi everyone and welcome to this lesson on how to find the volume of a solid of revolution by using what is called the disk method. So before we get started let me take you to a couple applets that will hopefully give you a visualization of what it is we're talking about here. The first one I want to share with you is off of the math demos.org website and if you take a look at the picture that's creating itself there. Let's get back to the beginning in a second. Notice that you start with a curve right there and imagine all those rings being akin to the Riemann sum idea of using inscribed rectangles under a curve. And remember with Riemann sums we talked about how if you increased the number of rectangles under a curve you get a better approximation of the actual area under a curve. That became the limit of a summation of those individual areas of those rectangles which then turned into a definite integral. That's what you're going to see here but now we're doing it with volume. So going back to the beginning with our curve all those rings represent all the representative rectangles. Think of it that way and as you revolve them around a axis of symmetry you're going to get a shape that is created. This is a really good website. I purposely left the URL up there so you could see it. I highly recommend you go to it and play with it. If you scroll down the page you'll notice they have some other you know lesson notes for you. They have some other visualizations some that you can even make yourself to envision better these volumes that we're creating and they give you a lot of very real-life examples of these. So I definitely encourage you to go to the website and maybe poke around a little bit. Another one I want to share with you is off of calculusapplets.com. So if you take a look at the picture there you can definitely see the outline of what looks to be a cylinder. The area that you see in yellow is the one that we're going to revolve around in this case the X axis. And notice how they drew for you one slice of the disc. Again think about the comparison to Riemann sums with finding area under a curve. So imagine having infinitely many of these discs inside the cylinder and if you found the volume of one individual disc and added them all up then you could approximate the volume of the solid that is created. Now if you look very carefully you're going to notice this dark gray rectangle. That's what I like to call the representative rectangle that you're going to be seeing in our lesson and examples. And imagine taking that representative rectangle and swinging it around the axis of revolution. That's what creates the disc. Now when I say disc think of like a frisbee or a coaster on which you would put a drink on a table. That's what we're talking about. So let's go back to the lesson and get started on how we're going to piece together a definite integral to actually find these volumes. So if you think about how to get volume of really any solid at all it's always going to be the area of the base multiplied by the height. Now in this case the height is pretty skinny because thinking again again of our representative rectangles. Remember how we talked about if you can make those rectangles really skinny you could fit more of them in there. So when we talk about height think maybe more of thickness and again it's not a really thick thickness. Alright so imagine something with a very very small height or thickness. So really what we're going to be talking about in order to get the volume of the solid that's created it is going to turn into a definite integral. So obviously we're going to have limits of integration. What we're taking the anti-derivative of the integral of is really the area of the base times the thickness. Now let's do our first example in terms of x. So this would be the area of the base and the dx is representative of the thickness. Remember dx is our differential that represents our delta x or change in x and an integral of course is the limit of a summation. So if you package this whole formula together hopefully it makes sense. What you're doing is really finding for each individual disk that you can imagine putting inside the solid you're going to find the area of the base multiplied by the thickness. Now sometimes you're going to find that our integral is going to be in terms of y instead. So perhaps that area of the base is in terms of y instead of x. So that becomes the basic setup of how we're going to find our volumes. So just to give you and clarify a definition of a solid of revolution it is a solid generated by revolving a region about a line and we call that line about which we are revolving the axis of revolution. Now most of the solids that we're going to be dealing with will involve a region bounded by a curve the x-axis and or the y-axis and perhaps some other horizontal or vertical line. So let's get started. So this is a visualization of something similar to the applet I took you to. So we have here a curve that's the dark black bold face curve you see here. In this case it is a form of a square root function and I'm going to take that shaded area the light gray region in between x equals a and x equals b and I'm going to revolve that around the x-axis. Now notice how inside that gray region I drew a representative rectangle. I really could have put that rectangle anywhere in that region. I could have drawn it over here. I could have drawn it over here. It literally just represents one that you would have. So I would take that dark black rectangle and start swinging it around my axis of revolution. Remember in this case our axis of revolution was the x-axis and the result when you do that is a disk. Think back to the applet visualizations I showed you. Now one thing to notice about our representative rectangle because this is going to lead to two characteristics of a disk method problem. Notice that representative rectangle is perpendicular to the axis of revolution and is also touching it. So imagine you had a whole lot of these disks in here and I wanted to find the volume of all of them add them up so as to approximate the volume of the solid. Well think we're talking disks. So the area of the base is pi r squared because it's circular and the height or thickness that's going to be either dx or dy because remember we're talking really skinny disks. Maybe a coaster, a flat coaster. You know the cardboard kind that you see in restaurants on which you put a drink. Think something like that. Now depending upon the graph that you have your representative rectangle could have different orientations and this is also connected to the type of axis of revolution you have. Is it either horizontal or vertical? Now if you have a horizontal axis of revolution such as the x-axis in the previous example, that radius that we need for the disk, notice I'm calling it capital R, it's going to be the distance from the axis of revolution to the curve. Now think in terms of how we've done this before. If you're trying to find the length of that representative rectangle, think top minus bottom. The thickness, the height, is going to be represented by dx and that's because it's a vertical representative rectangle. So the way in which you're going to get your volume then is you're going to do the integral from A to B and they have to be x values of pi multiplied by the radius squared. So the capital R it has to be in terms of x since it's a dx problem and to get that r value you're going to have an expression composed of the top of the rectangle and what that hits minus the bottom of the rectangle. Now if by chance your solid is revolved about a vertical axis now you're going to have a representative rectangle that is oriented horizontally that makes it a dy problem. The r value is going to still be the length of your representative rectangle from the axis of revolution to the curve. Now this time you're going to think right minus left. So the r value and again it's going to be an expression that you'll compose by taking what the representative rectangle is hitting on the right side minus what it's hitting on the left side. Your volume then is going to be the integral from C to D. They need to be y value since this is a dy problem of pi times that r expression you got that quantity squared. The evaluation you'll typically simply do in your graphing calculator.