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

#### MarkFL

Staff member
Here is the question:

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.

#### MarkFL

Staff member
Hello Emily,

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

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:

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