Volume by Disc and Shell Methods
6.4 Volume by shells
Note: Section numbers refers to Calculus, Early Transcendentals (2nd edition) by Briggs, Cochran, and Gillett.
These prints illustrate the disc method and the shell method for finding volumes of solids of revolution.
The blue object is obtained by revolving the region between y-axis and the graph of:
The model on the left shows the solid. The one on the right is an approximation using 10 shells.
In the solid below (titled the Volumes of Hanoi) the piece on the left of the model is a solid of revolution obtained by rotating the region bounded by the graph of the function
The rightmost set of pieces represents an approximation of volume of the center solid of revolution with four discs, and the middle set represents a different approximation of that solid with four shells. The discs each have the same height but their radii are determined by the function f(x). The shells each have the same thickness but their heights are determined by the function f(x).
On one side is the approximation of the volume using four disks, and on the other side is the approximation of the volume using four shells. To make it clear which is which, we have shown the same model with the pieces taken apart.
The next model is a solid obtained by revolving the region between the graph of
The model on the left is the exact region, and the one on the right is an approximation which uses 8 shells.
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