Wednesday, August 24, 2011

Gabriel’s Horn

Gabriel’s Horn is a three dimensional surface that contains a finite volume but has an infinite surface area. It is made by taking the two dimensional graph of y=1/x and revolving it around the x-axis (with the domain of x ≥ 1). If we look at x coordinates from 1 to a, the volume can be found by the equation: Which is really just sum of the area of each circular cross section, hence it is the integral of πr2 (with r being the distance from the x-axis to the function). The surface area of the horn can be found by the equation: (Which is slightly more complicated to derive but a full explanation can be found here.) As you can see if we let a approach infinity, the surface area diverges, whereas the volume converges to π.

Posted at 11:29 PM 47 notes Tags: math geometry gabriel's horn infinity volume area

Notes

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    LOL I love calculus. Except I don’t understand
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    area… seems pretty magical no?
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