# Volume of solid of revolution

• Mar 13th 2010, 11:09 AM
jupiter92
Volume of solid of revolution
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.
• Mar 13th 2010, 11:24 AM
chisigma
The requested volume is...

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Mar 13th 2010, 11:30 AM
jupiter92
thank you,could you tell me where did you get that e^2x?
• Mar 13th 2010, 11:32 AM
TheMasterMind
Quote:

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= $\displaystyle 2\pi x(1-ln x)$

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

$\displaystyle \int^{e}_{1} 2\pi x (1- ln x) dx$

now integrate
• Mar 13th 2010, 11:45 AM
chisigma
Quote:

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 $\displaystyle 0\le x \le 1$ and $\displaystyle 1\le y \le e^{x}$ and rotation is around the x axis. This conducts to use the formula...

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Mar 13th 2010, 12:06 PM
jupiter92
thanks.
• Mar 13th 2010, 12:07 PM
jupiter92
btw,could you tell me what does the dx stand for?