(New page: Suppose that <math>f(x)</math> is a continuous function on the interval <math>[a,b]</math>. Let <math>V(c)</math> denote the volume of the solid obtained by revolving the area between the ...) |
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Revision as of 15:08, 14 September 2008
Suppose that $ f(x) $ is a continuous function on the interval $ [a,b] $. Let $ V(c) $ denote the volume of the solid obtained by revolving the area between the graph of $ y=f(x) $ and the line $ y=c $ about the line $ y=c $.
As $ c $ ranges from $ m=\{\text{min }f(x):a\le x\le b\} $ and $ M=\{\text{Max }f(x):a\le x\le b\} $, prove that the minmum value of $ V(c) $ is attained at $ c $ equal to the average value of $ f(x) $ on the interval $ [a,b] $.
