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<math>V(c)= \int_a^b \pi (f(x)-c)^2\ dx = Ac^2-Bc+D</math> | <math>V(c)= \int_a^b \pi (f(x)-c)^2\ dx = Ac^2-Bc+D</math> | ||
| − | where <math> | + | where <math>D=\pi \int_a^b f(x)^2\ dx</math> is a constant which is positive if <math>f</math> |
| − | is not identically zero, <math>B=2\pi\int_a^b f(x)\ dx</math> and <math> | + | is not identically zero, <math>B=2\pi\int_a^b f(x)\ dx</math> and <math>A=\pi(b-a)</math>. |
The case where <math>f(x)</math> is identically zero is easy. The minimum volume is zero | The case where <math>f(x)</math> is identically zero is easy. The minimum volume is zero | ||
and occurs at <math>c=0</math>, which is the average value of <math>f</math>. | and occurs at <math>c=0</math>, which is the average value of <math>f</math>. | ||
| − | Hence, we may assume that <math>f</math> is not identically zero and that <math> | + | Hence, we may assume that <math>f</math> is not identically zero and that <math>D</math> is |
a positive constant. In this case, the graph of <math>V(c)</math> is an upward opening | a positive constant. In this case, the graph of <math>V(c)</math> is an upward opening | ||
| − | parabola. | + | parabola. Such a parabola has an absolute minimum at the value <math>c</math> such that |
| + | <math>V'(c)=0</math>. Since <math>V'(c)=2Ac-B</math>, this value of <math>c</math> is | ||
| + | |||
| + | <math>c= \frac{B}{2A} = \frac{1}{b-a}\int_a^b f(x)\ dx,</math> | ||
| + | |||
| + | which is the average value of <math>f</math> on the interval. To complete the argument, | ||
| + | we must show that this value of <math>c</math> falls in the interval between | ||
| + | <math>m</math> and <math>M</math>, and this is easy because | ||
| + | |||
| + | <math>m \le f(x) \le M</math> | ||
| + | |||
| + | on <math>[a,b]</math> implies that ... | ||
Revision as of 08:15, 29 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] $.
I will give Josh Hunsberger's proof here.
Proof. Notice that
$ V(c)= \int_a^b \pi (f(x)-c)^2\ dx = Ac^2-Bc+D $
where $ D=\pi \int_a^b f(x)^2\ dx $ is a constant which is positive if $ f $ is not identically zero, $ B=2\pi\int_a^b f(x)\ dx $ and $ A=\pi(b-a) $.
The case where $ f(x) $ is identically zero is easy. The minimum volume is zero and occurs at $ c=0 $, which is the average value of $ f $.
Hence, we may assume that $ f $ is not identically zero and that $ D $ is a positive constant. In this case, the graph of $ V(c) $ is an upward opening parabola. Such a parabola has an absolute minimum at the value $ c $ such that $ V'(c)=0 $. Since $ V'(c)=2Ac-B $, this value of $ c $ is
$ c= \frac{B}{2A} = \frac{1}{b-a}\int_a^b f(x)\ dx, $
which is the average value of $ f $ on the interval. To complete the argument, we must show that this value of $ c $ falls in the interval between $ m $ and $ M $, and this is easy because
$ m \le f(x) \le M $
on $ [a,b] $ implies that ...
