Volume of a solid involving natural logs

• Feb 20th 2009, 01:45 PM
fattydq
Volume of a solid involving natural logs
I'm trying to solve the problem I've attached as a picture. I'm having some difficulty involving the step where I get to the integral from 1 to 3 of pi times (ln(4x)^2, because I don't really even know what that is. I thought it'd just be ln^2(4x) but then I have NO idea how to integrate that! I'd appreciate anyone who's willing to help.
• Feb 20th 2009, 02:48 PM
mollymcf2009
Quote:

Originally Posted by fattydq
I'm trying to solve the problem I've attached as a picture. I'm having some difficulty involving the step where I get to the integral from 1 to 3 of pi times (ln(4x)^2, because I don't really even know what that is. I thought it'd just be ln^2(4x) but then I have NO idea how to integrate that! I'd appreciate anyone who's willing to help.

First, remember that since you are using disks or washers for this problem and rotating about the y axis, your integral will need to be in terms of y.

$ln(4x) = y$
$e^y = 4x$
$x = \frac{e^y}{4}$

$\int \pi (\frac{e^y}{4})^2 dy$
$2{\pi}\left[\int_{0}^{\frac{e^{3}}{4}}x(3-ln(4x))-\int_{0}^{\frac{e}{4}}x(1-ln(4x))\right]dx$