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AP Calculus Review Disk and Washer Methods

The disk and washer methods are useful for finding volumes of solids of revolution. In this article, well review the methods and work out a number of example problems. By the end, youll be prepared for any disk and washer methods problems you encounter on the AP Calculus AB/BC exam! Solids of Revolution The disk and washer methods are specialized tools for finding volumes of certain kinds of solids solids of revolution. So what is a solid of revolution? Starting with a flat region of the plane, generate the solid that would be swept out as that region revolves around a fixed axis. For example, if you start with a right triangle, and then revolve it around a vertical axis through its upright leg, then you get a cone. The cone generated as a solid of revolution by revolving a right triangle around a vertical axis Heres another cool example of a solid of revolution that you might have seen hanging up as a decoration! Tissue paper decorations that unfold from flat to round are examples of solids of revolution. Watch the next few seconds of the video below to see how it unfolds in real time. The Disk and Washer Methods: Formulas So now that you know a bit more about solids of revolution, lets talk about their volumes. Suppose S is a solid of revolution generated by a region R in the plane. There are two related formulas, depending on how complicated the region R is. Disk Method The simplest case is when R is the area under a curve y = f(x) between x = a and x = b, revolved around the x-axis. Now imagine cutting the solid into thin slices perpendicular to the x-axis. Each slice looks like a disk or cylinder, except that the outer surface of the disk may have a curve or slant. Lets approximate each slice by a cylinder of height dx, where dx is very small. In fact, I like to think of each disk as being generated by revolving a thin rectangle around the x-axis. Then you can see that the height of the rectangle, y, is the same as the radius of the disk. Now lets compute the volume of a typical disk located at position x. The radius is y, which itself is just the function value at x. That is, r = y = f(x). The height of the disk is equal to dx (think of the disk as a cylinder standing on edge). Therefore, the volume of a single cylindrical disk is: V = Ï€ r2h = Ï€ f(x)2dx. This calculation gives the approximate volume of a thin slice of S. Next, to approximate the volume of the entirety of S, we have to add up all of the disk volumes throughout the solid. For simplicity, assume that the thickness of each slice is constant (dx). Also, for technical reasons, we have to keep track of the various x-values along the interval from a to b using the notation xk for a generic sample point. Finally, by letting the number of slices go to infinity (by taking a limit as n → ∞), we develop a useful formula for volume as an integral. Example 1: Disk Method Let R be the region under the curve y = 2x3/2 between x = 0 and x = 4. Find the volume of the solid of revolution generated by revolving R around the x-axis. Solution Lets set up the disk method for this problem. The volume of the solid is 256Ï€ (roughly 804.25) cubic units. Washer Method Now suppose the generating region R is bounded by two functions, y = f(x) on the top and y = g(x) on the bottom. This time, when you revolve R around an axis, the slices perpendicular to that axis will look like washers. No, were not talking about clothes washers or dishwashers A washer is like a disk but with a center hole cut out. The formula for the volume of a washer requires both an inner radius r1 and outer radius r2. Well need to know the volume formula for a single washer. V = Ï€ (r22 r12) h = Ï€ (f(x)2 g(x)2) dx. As before, the exact volume formula arises from taking the limit as the number of slices becomes infinite. Example 2: Washer Method Determine the volume of the solid. Here, the bounding curves for the generating region are outlined in red. The top curve is y = x and the bottom one is y = x2 Solution This is definitely a solid of revolution. Well set up the formula with f(x) = x (top) and g(x) = x2 (bottom). But what should we use as a and b? Well, just as in some area problems, you may have to solve for the bounds. Clearly the region is bounded by the two curves between their common intersection points. Set f(x) equal to g(x) and solve to locate these points of intersection. x = x2   Ã‚  Ã¢â€ â€™Ã‚  Ã‚   x x2 = 0   Ã‚  Ã¢â€ â€™Ã‚  Ã‚   x(1 x) = 0. We find two such points: x = 0 and 1. So set a = 0 and b = 1 in the formula. Example 3: Different Axes Set up an integral that computes the volume of the solid generated by revolving he region bounded by the curves y = x2 and x = y3 around the line x = -1. Solution Be careful not to blindly apply the formula without analyzing the situation first! This time, the axis of rotation is a vertical line x = -1 (rather than the horizontal x-axis). The radii will be horizontal segments, so think of x1 and x2 (rather than y-values). Furthermore, because everything is turned on its side compared to previous problems, we have to make sure both boundary functions are solved for x. The thickness of the washer is now dy (instead of dx). Finally, because the axis of revolution is one unit to the left of the y-axis, that adds another unit to each radius. (The further away the axis, the longer the radius must be to reach the figure, right?) Take a look at the graph below to help visualize whats going on. Inner Radius: x = y3 + 1 Outer Radius: x = y1/2 + 1 As before, set the functions equal and solve for points of intersection. Those are again at x = 0 and 1. Using the Washer Method formula for volume, we obtain: The problem only asks for setup, so we are done at this point.

How Do You Write a College Essay About Typing

How Do You Write a College Essay About Typing?How do you write a college essay about typing? You have probably spent a good portion of your life writing essays. Are you aware that the easiest way to write a college essay about typing is to use the same framework you use for all other essays?To write an essay about typing, you need to bring out the essence of your essay. It is not enough to simply describe the act of typing. Instead, you have to bring out the ideas from your essay and make them more interesting and vivid by adding some minor details.In writing an essay about typing, it is important to take into account that there are three types of questions you can expect to be asked in your essay. The first question would be the actual question, 'What is typing?' The second question would be, 'How did you develop your typing technique?' And finally, there would be the question, 'What are the reasons why you decided to switch careers?'The first question usually will be the subject of the essay itself. So, instead of just writing a single paragraph that would be the topic of the entire essay, you have to break up the whole essay into many paragraphs. This would make it easier for the reader to follow and understand. The second question will be the theme of the entire essay.The main reason why people switch careers is because they found a better-paying and more exciting job. They can often also choose a career that is much less stressful. Before writing your essay, you need to learn as much as you can about the changes in technology.When you are writing an essay about typing, the topic should be something that is easily understandable to the reader. The use of bold fonts would be helpful in this situation. Using color would also be helpful in drawing the reader's attention. Bold colors are a good way to draw the reader's attention to an important point.Finally, using short sentences would also help make your essay short and easy to read. It would be best to try t o make your essay about typing as short as possible. It would be a lot easier for the reader to remember and understand if it is short and easy to read.As you can see, how do you write a college essay about typing is really very simple. All you need to do is use the same information that you use for all other essays. Using this framework, it would be very easy for you to create your own special college essay about typing.