Problem:
Graphically illustrate the approximation of the volume


[Press this to see MathView Notebook.]

which is obtained by rotating the area bounded by the graph of

f(x) = x - x2

and the x-axis about the x-axis.

Visualization:


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In the animation above, you can see how by increasing the number of equal-sized intervals the sum of the volumes of cylinders can better approximate the volume of the rotated solid.

The following table indicates the sums of the various volumes of the cylinders n indicates the number of cylinders.

nSum of volumes of
cylinders
20.11044662
40.10507768
80.10474213
160.10472115
320.10471984
640.10471976
1280.10471976

0.10471976 appears to be a reasonable estimate of our volume.