# Thread: [SOLVED] area infinite, volume finite

1. ## [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?

2. 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.

3. 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.

4. 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!

5. God, that is infinitely weirder.