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Math Help - [SOLVED] area infinite, volume finite

  1. #1
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    [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?
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by diroga View Post
    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.
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  3. #3
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    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.
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  4. #4
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    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!
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  5. #5
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    God, that is infinitely weirder.
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