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Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 7.3 day 2 Disk and Washer Methods Limerick Nuclear Generating Station, Pottstown, Pennsylvania

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Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape. So I put a piece of wood in a lathe and turn it to a shape to match the curve.

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How could we find the volume of the cone? One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes. The volume of each flat cylinder (disk) is: In this case: r= the y value of the function thickness = a small change in x = dx

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The volume of each flat cylinder (disk) is: If we add the volumes, we get:

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This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk. If the shape is rotated about the x-axis, then the formula is: Since we will be using the disk method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes. A shape rotated about the y-axis would be:

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The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. y x We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk

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The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

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The region bounded by and is revolved about the y-axis. Find the volume. The “disk” now has a hole in it, making it a “washer”. If we use a horizontal slice: The volume of the washer is: outer radius inner radius

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This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle. The washer method formula is: Like the disk method, this formula will not be on the formula quizzes. I want you to understand the formula.

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If the same region is rotated about the line x = 2 : The outer radius is: R The inner radius is: r

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Volume is understood as length times width times height, or on a graph, x times y times z. When an integral is revolved around an axis, this is the area.

Volume is understood as length times width times height, or on a graph, x times y times z. When an integral is revolved around an axis, this is the area.

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