Thread: Volume of solid of revolution

1. 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.

2. 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

$\displaystyle \chi$ $\displaystyle \sigma$

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

4. 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

5. 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

$\displaystyle \chi$ $\displaystyle \sigma$

6. thanks.

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

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