Calc07_3day2 - 7.3 day 2 Disks Washers and Shells Limerick...

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Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003 7.3 day 2 Disks, Washers and Shells Limerick Nuclear Generating Station, Pottstown, Pennsylvania
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y x = 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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y x = 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: 2 the thickness r π In this case: r= the y value of the function thickness = a small change in x = dx π ( 29 2 x dx
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y x = The volume of each flat cylinder (disk) is: 2 the thickness r π If we add the volumes, we get: ( 29 2 4 0 x dx π 4 0 x dx π = 4 2 0 2 x π = 8 π = π ( 29 2 x dx
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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: 2 b a V y dx π = 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. 2 b a V x dy π = 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.
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