Gabriel's horn is a solid of revolution that has a finite volume but an infinite surface area.pic.twitter.com/p9xYUpvJ2B
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No, I meant aside from the typo, you are computing volume in [1,\infty),not (0,\infty) where BOTH integrals are not convergent.
I don't see what the issue is.. Why would I compare two divergent quantities? The volume integral converges for every ε>0 , and the area integral diverges independent of starting point. We could start both from 0 if we just integrated 1/(x+1)^2 instead and the result is the same
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\int_ {\epsilon}^{\infty}\frac {1} {x^2}\, dx = \frac {1} {\epsilon} which goes to infinity
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So it is the artifact of x>1 that makes the integral converge.
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And it is also infinity minus minus infinity case which gives infinity again.
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