Shells Notes - the curve y = 2 x-x 2 and the x-axis about...

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Review: Volume by slicing: Formula: V = Z b a A ( x ) dx where A ( x ) is the area of the cross section. When do I use this formula? Volumes of Solids of Revolution - Disk/Washer Method: Formula When to use : V = Z b a π ( R ( x )) 2 dx V = Z d c π ( R ( y )) 2 dy V = Z b a π ± ( R ( x )) 2 - ( r ( x )) 2 ² dx V = Z d c π ± ( R ( y )) 2 - ( r ( y )) 2 ² dy
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Volumes of Solids of Revolution - Cylindrical Shells Motivating example Find the volume of the solid generated by rotating the region bounded by the curve y = 2 x - x 2 and the x-axis about the y-axis. Why can’t we just use the disk/washer method? Cylindrical Shells: Consider representative strips which are vertical to the axis of rotation. What shape does the rotation of a strip make? What is the volume of this strip?
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If we add all the volumes of the cylinders together? To summarize, when using the method of cylindrical shells, the formulas are: V = Z b a 2 π · h ( x ) · r ( x ) dx when rotating around a vertical line, and V = Z d c 2 π · h ( y ) · r ( y ) dy when rotating around a horizontal line. h is the height of the cylinder and r is the radius. Back to the motivating example: Find the volume of the solid generated by rotating the region bounded by
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Unformatted text preview: the curve y = 2 x-x 2 and the x-axis about the y-axis. Rotating about the x-axis: Consider the region in the rst quadrant bounded by the curves y 2 = x and y = x 3 . Use the shell method to compute the volume of the solid obtained by rotating this region around the x-axis. On your own: Consider the region in the rst quadrant bounded by the curves y 2 = x and y = x 3 . Use the shell method to compute the volume of the solid obtained by rotating this region around the y-axis. Rotating about lines other than the axes: 1. Consider the region bounded by the graphs of y = x 2 and y = x + 2. Use the method of shells to nd the volume of the solid obtained by rotating about the line x = 3. 2. Find the volume of the solid obtained by rotating the region bounded by y = x-x 2 and the x axis about the line x = 2....
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This note was uploaded on 10/02/2011 for the course MATH 2214 at Virginia Tech.

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Shells Notes - the curve y = 2 x-x 2 and the x-axis about...

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