I am no specialist in integration but I would like to know what is the (arc?) length of the function (along the function) between $f(a)$ and $f(b)$ where $ f(x) = \sqrt{(x^2+1)} $
I would like to see the deduction.
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I think one usually writes $L=\int_a^bds$, where $ds$ satisfies $ds^2=dx^2+dy^2$, so $ds =\sqrt{dx^2+dy^2}$, i.e., $ds=\sqrt{1+(\frac{dy}{dx})^2}$. So plug in your function and compute the derivative etc. and try to evaluate. These sorts if integrals aren't always evaluable directly and may require numerical approximation. – MPW May 04 '14 at 11:51
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I know you ask deep questions. I'm still searching for an upper bound. Then we can use pinch theorom to complete the "deduction." Yes, I think the def alone is inadequate. – George Chen Sep 09 '14 at 13:04