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Math Help - Rotated infinite region

  1. #1
    Jan 2013

    Rotated infinite region

    Rotated infinite region-image.jpgFor this problem I started with part a by saying that the area diverges by using the p test where p=1. For the second half I found the area of a cross sectional washer to be pi(1 - (1/x^2))delta x. Taking the integral of this from to infinity would result in an infinitesimally large area - an infinitesimally small area, thus resulting in another divergence. Is this correct? Why should I have been surprised by the answer?
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  2. #2
    Senior Member
    Feb 2013
    Saudi Arabia

    Re: Rotated infinite region


    integrate from 1 to infinity the function 1/x to obtain +infinity .this means that surprisingly the area enclosed by the graph 1/x and the x-axis from 1 to infinity is extremely large though it is not seem to be so!. The curve 1/x aproaches the is an asymptote and they meet at a point very far away...infinity.

    Rotating the graph of curve y=1/x over the x-axis you have to integrate the function π(1/x)^2 from 1 to infinity but surprisingly you will find the volume to be equal to π

    This is the surprise....the curve has an infinite area but if rotated it generates a finite volume...

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