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7.3 Volumes Disk and Washer Methods

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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: 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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Example: Using calculus, derive the formula for finding the volume of a sphere of radius r. The region bounded by a semicircle and its diameter shown below is revolved about the x-axis, which gives us a sphere of radius r. dx Area of each cross section? (circle) What is y? r–r Equation of semicircle:

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Example: Using calculus, derive the formula for finding the volume of a sphere of radius r. Area of each cross section? (circle) r–r Volume: (Remember r is just a number!!!)

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Example: Using calculus, derive the formula for finding the volume of a sphere of radius r. r–r Volume:

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Example: Find the volume of the solid generated when the region bounded by y = x 2, x = 2, and y = 0 is rotated about the line x = 2. dy Area of each cross section? (circle) What is r? r 2 r = 2 – x Remember, we are using a dy here!!! So,

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Example: Find the volume of the solid generated when the region bounded by y = x 2, x = 2, and y = 0 is rotated about the line x = 2. Area of each cross section? (circle) r 2 Volume: Bounds? From 0 to intersection (y-value!!!)

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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:

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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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