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Published byKristina Carroll Modified over 8 years ago

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Review: Volumes of Revolution

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x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge shape several ways, but the simplest cross section is a rectangle. If we let h equal the height of the slice then the volume of the slice is: Since the wedge is cut at a 45 o angle: x h 45 o Since

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x y Even though we started with a cylinder, does not enter the calculation!

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Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections

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7.3 Disk and Washer Methods

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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:

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Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

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Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

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7.3 The Shell Method

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Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

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Shell method: If we take a vertical slice and revolve it about the y-axis we get a cylinder.

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Note:When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

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When the strip is parallel to the axis of rotation, use the shell method. When the strip is perpendicular to the axis of rotation, use the washer method.

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Volumes Using Cylindrical Shells The region bounded by the curve y =, the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid using cylindrical shells.

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Volumes Using Cylindrical Shells The region bounded by the curve y =, the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid using cylindrical shells.

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