Volume of solid of revolution

**jupiter92** · March 13th 2010, 11:09 AM

Let R be the region in Quadrant 1 bounded by the graphs of y=lnx and y=1.Find the exact volume of the solid generated by rotating R about the y-axis.

Thanks for help.

**chisigma** · March 13th 2010, 11:24 AM

The requested volume is...

$V= \pi\cdot \int_{0}^{1} e^{2x}\cdot dx = \frac{\pi}{2}\cdot (e^{2}-1)$ (1)

Kind regards

$\chi$ $\sigma$

**jupiter92** · March 13th 2010, 11:30 AM

thank you,could you tell me where did you get that e^2x?

**TheMasterMind** · March 13th 2010, 11:32 AM

Originally Posted by jupiter92

Let R be the region in Quadrant 1 bounded by the graphs of y=lnx and y=1.Find the exact volume of the solid generated by rotating R about the y-axis.

Thanks for help.

A=circumference x height= $2\pi x(1-ln x)$

$dV= 2\pi x (1- ln x) dx$

$<br /> \int^{e}_{1} 2\pi x (1- ln x) dx$

now integrate

**chisigma** · March 13th 2010, 11:45 AM

Originally Posted by jupiter92

thank you,could you tell me where did you get that e^2x?

Simply I swap the y-axis and the x-axis so that the area is defined for $0\le x \le 1$ and $1\le y \le e^{x}$ and rotation is around the x axis. This conducts to use the formula...

$V= \pi\cdot \int_{0}^{1} f^{2}(x)\cdot dx$ (1)

... where $f(x)= e^{x}$ , formula that leads to a more confortable integration...

Kind regards

$\chi$ $\sigma$

**jupiter92** · March 13th 2010, 12:06 PM

thanks.

**jupiter92** · March 13th 2010, 12:07 PM

btw,could you tell me what does the dx stand for?

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