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

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The discussion revolves around calculating the volume of a solid formed by rotating a region bounded by the curves y=lnx, y=1, y=2, and x=0 about the y-axis. The volume is derived using the disk method, leading to the expression dV=πr²dy, where r is defined as x=e^y. The integral for the volume is formulated as V=π∫₁² e²ʸ dy, which is then simplified through substitution to V=(π/2)∫₂⁴ eᵘ du. Finally, applying the Fundamental Theorem of Calculus yields the volume as V=(π/2)(e⁴-e²).
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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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Hello Emily,

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

View attachment 1666

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

$$dV=\pi r^2\,dy$$

where:

$$y=x=e^y$$

Hence:

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

Summing all the disks through integration, we may write:

$$V=\pi\int_1^2 e^{2y}\,dy$$

If we use the substitution:

$$u=2y\,\therefore\,du=2\,dy$$

we may write:

$$V=\frac{\pi}{2}\int_2^4 e^{u}\,du$$

Applying the FTOC, we find:

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

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