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Sparkling's question at Yahoo! Answers regarding a volume by slicing

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MarkFL

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Feb 24, 2012
13,775
Here is the question:

How to find the volume of a solid given an equation and bounds?

Let the first quadrant region enclosed by the graph of =1/x, x=1 and x=4 be the base of a solid. If cross sections perpendicular to the x-axis are semicircles, the volume of the solid is:

(Answer: 3pi/ 32)

Please tell me how you got the answer because I kept getting 3pi/4
I have posted a link there to this thread so the OP can view my work.
 
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MarkFL

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Feb 24, 2012
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Hello Sparkling,

The volume of an arbitrary semicircular slice is:

\(\displaystyle dV=\frac{\pi}{8}D^2\,dx\)

where the diameter $D$ is:

\(\displaystyle D=\frac{1}{x}\)

hence:

\(\displaystyle dV=\frac{\pi}{8}x^{-2}\,dx\)

And so, the sum of all the slices is given by:

\(\displaystyle V=\frac{\pi}{8}\int_1^4 x^{-2}\,dx\)

Applying the FTOC, we obtain:

\(\displaystyle V=\frac{\pi}{8}\left[-\frac{1}{x} \right]_1^4=\frac{\pi}{8}\left(1-\frac{1}{4} \right)=\frac{3\pi}{32}\)