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Emily's questions at Yahoo! Answers regarding a solid of revolution

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MarkFL

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Feb 24, 2012
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Here is the question:

Please help with this simple integral question, thank you?


Write the integral for the volume of the solid obtained by rotating the region bounded by y=lnx, y=1, y=2, and x=0 about the y axis.


THANK YOU SO MUCH!
I have posted a link there to this thread so the OP can see my work.
 
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MarkFL

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Feb 24, 2012
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Hello Emily,

First, let's plot the region to be revolved:

emily.jpg

Using the disk method, we may write the volume of an arbitrary disk as follows:

\(\displaystyle dV=\pi r^2\,dy\)

where:

\(\displaystyle y=x=e^y\)

Hence:

\(\displaystyle dV=\pi \left(e^y \right)^2\,dy=\pi e^{2y}\,dy\)

Summing all the disks through integration, we may write:

\(\displaystyle V=\pi\int_1^2 e^{2y}\,dy\)

If we use the substitution:

\(\displaystyle u=2y\,\therefore\,du=2\,dy\)

we may write:

\(\displaystyle V=\frac{\pi}{2}\int_2^4 e^{u}\,du\)

Applying the FTOC, we find:

\(\displaystyle V=\frac{\pi}{2}\left[e^u \right]_2^4=\frac{\pi}{2}\left(e^4-e^2 \right)\)