Chris Nash wrote:
>If y=f(x), the volume of revolution is given by
{snip}
Chris, you forgot to explicitly give that f(x) = 1/x, although one
can infer that from the rest of your psot.
Brain teaser:
Are there infinitely many smooth functions f(x) which have the same
volume as the classical Gabriel's Horn when rotated about the x-axis
and x goes from 1 to infinity, and which also have infinite surface area?
(It would suffice to find a one-parameter function family which satisfies
this criterion.)
Gabriel's Horn is one of my favorite Calculus problems. Another is
the famous closed-form method of finding the integral of the Gaussian
exp(-x^2), where the fascinating number sqrt(pi) appears in the result.
The famous Euler identity, exp(i*pi)+1=0, which can be obtained from
the Taylor series for the exponential function and which involves the
five most fundamental constants in mathematics, is yet another.
Gauss' "Theorema Egregium" ("Totally awesome theorem" in dude-speak :)
about the curvature of surfaces also deserves mention, but is getting
way off-topic.
Cheers,
-Ernst
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