Symmetric arc length of ln(x) and e^x

[icesalmon]

Homework Statement

Explain why ∫(1+(1/x²)^1/2dx over [1,e] = ∫(1+e^2x)^1/2dx over [0,1]

The Attempt at a Solution

The two original functions are ln(x) and e^x and are both symmetrical about the line y = x. If I take either of the functions and translate it over the line y = x the two functions will match up completely. So it seems reasonable that the arc lengths will be the same over some region. If I plug in the bounds 1 and e into ln(x) i get 0, and 1 and if I plug the bounds 0,1 into e^x I get 1, and e. I don't really know how it helps but it's something I suppose.

 

[haruspex]

Your argument appears sound. Are you looking for a more algebraic justification? It shouldn't be too hard to turn your argument into algebra.

 

[icesalmon]

I am looking for a more algebraic justification. I'll try and clean it up and post back when I have something, or if I have any questions. Thanks.