# [SOLVED] area infinite, volume finite

• June 13th 2009, 06:29 PM
diroga
[SOLVED] area infinite, volume finite
The region P is bounded by the x axis , the line x =1 and the curve y = 1/x. show that
a. the area of region P is infinite.
b. show that the solid of revolution obtained by rotating region P about the x axis is finite.

a. $A = \int_{1}^{\infty}\frac{1}{x} dx = \lim_{t \rightarrow \infty} \ln (x) \mid_{1}^{t} = \infty - \ln(1)$ diverges

b. $V = \pi \int_{1}^{\infty} \frac{1}{x^2} dx = \lim_{t \rightarrow \infty} \pi(-\frac{1}{x}) \mid_{1}^{t} = \pi(0 + 1) = \pi$

Am I correct?
• June 13th 2009, 06:41 PM
apcalculus
Quote:

Originally Posted by diroga
The region P is bounded by the x axis , the line x =1 and the curve y = 1/x. show that
a. the area of region P is infinite.
b. show that the solid of revolution obtained by rotating region P about the x axis is finite.

a. $A = \int_{1}^{\infty}\frac{1}{x} dx = \lim_{t \rightarrow \infty} \ln (x) \mid_{1}^{t} = \infty - \ln(1)$ diverges

b. $V = \pi \int_{1}^{\infty} \frac{1}{x^2} dx = \lim_{t \rightarrow \infty} \pi(-\frac{1}{x}) \mid_{1}^{t} = \pi(0 + 1) = \pi$

Am I correct?

I don't see anything wrong with your work on this problem.
• June 14th 2009, 06:55 AM
pomp
Yes this is perfectly correct, the shape is know as Gabriel's Horn and the problem is sometimes reffered to as The Painter's Paradox.

Gabriel's Horn - Wikipedia, the free encyclopedia

It's quite weird that you can fill the horn to the brim with a finite amount of paint, but to paint the outside you need an infinite amount.

pomp.
• June 14th 2009, 12:48 PM
HallsofIvy
Actually, it's wierder than that! You can fill it to the brim with a finite amount of paint but the inside surface cannot be covered by paint!
• June 15th 2009, 03:08 AM
pomp
(Surprised) God, that is infinitely weirder.