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6.3 Volumes by Cylindrical Shells
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Find the volume of the solid obtained by rotating the region bounded,, and about the y -axis. We can use the washer method if we split it into two parts: outer radius inner radius thickness of slice cylinder Japanese Spider Crab Georgia Aquarium, Atlanta Example
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cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx. If we take a vertical slice and revolve it about the y-axis, we get a cylinder. If we add all of the cylinders together, we can reconstruct the original object.
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If we add all the cylinders from the smallest to the largest: This is called the shell method because we use cylindrical shells.
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For vertical axis of revolution, the volume is For horizontal axis of revolution, the volume is The Shell Method
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If we take a vertical slice and revolve it about the y-axis, we get a cylinder. Find the volume generated when this shape is revolved about the y axis. Example
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1)Determine whether a horizontal or vertical strip should be used (vertices of the representative rectangle should lie on two different graphs on the entire interval of interest.) 2)Determine which method to be used: o If the strip is perpendicular to the axis of rotation, use the washer method or disk method. o If the strip is parallel to the axis of rotation, use the shell method. Disk Method vs Shell Method
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Find the volume of the solid of revolution generated by rotating the curve y = x 2, x = 0 and y = 4 about the given line. 1) y-axis 2) x-axis 3) the line x = 3 4) the line y = - 2 Examples
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