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Lesson 7.3 (1) - Volumes of Revolution The Shell Method...

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Volumes of Revolution The Shell Method Lesson 7.3
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2 Shell Method Based on finding volume of cylindrical shells Add these volumes to get the total volume Dimensions of the shell Radius of the shell Thickness of the shell Height
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3 The Shell Consider the shell as one of many of a solid of revolution The volume of the solid made of the sum of the shells f(x) g(x) x f(x) – g(x) dx [ ] 2 ( ) ( ) b a V x f x g x dx π = -
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4 Try It Out! Consider the region bounded by x = 0, y = 0, and 2 8 y x = - 2 2 2 0 2 8 V x x dx π = -
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5 Hints for Shell Method Sketch the graph over the limits of integration Draw a typical shell parallel to the axis of revolution Determine radius, height, thickness of shell Volume of typical shell Use integration formula 2 radius height thickness π 2 b a Volume radius height thickness π =
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6 Rotation About x-Axis Rotate the region bounded by y = 4x and y = x 2 about the x-axis What are the dimensions needed? radius height
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