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Shells Notes

# 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?
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 ﬁrst 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 ﬁrst 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 ﬁnd 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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Shells Notes - the curve y = 2 x-x 2 and the x-axis about...

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