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

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SUMMARY

The discussion centers on proving the equality of the arc lengths of the functions ln(x) and e^x over specified intervals. Specifically, it establishes that ∫(1+(1/x²)^(1/2)dx from 1 to e equals ∫(1+e^(2x))^(1/2)dx from 0 to 1 due to the symmetry of the functions about the line y = x. The participants agree that translating either function across this line results in identical shapes, thereby justifying the equality of their arc lengths. Further algebraic justification is sought to solidify this conclusion.

PREREQUISITES
  • Understanding of integral calculus, specifically arc length calculations.
  • Familiarity with the properties of logarithmic and exponential functions.
  • Knowledge of symmetry in mathematical functions.
  • Ability to manipulate and simplify algebraic expressions.
NEXT STEPS
  • Study the derivation of arc length formulas for both ln(x) and e^x.
  • Explore the concept of function symmetry and its implications in calculus.
  • Learn about the application of definite integrals in calculating arc lengths.
  • Investigate algebraic techniques for transforming functions to demonstrate equivalence.
USEFUL FOR

Students in calculus courses, mathematics educators, and anyone interested in understanding the properties of logarithmic and exponential functions, particularly in relation to arc length calculations.

icesalmon
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Homework Statement


Explain why ∫(1+(1/x2)1/2dx over [1,e] = ∫(1+e2x)1/2dx over [0,1]

The Attempt at a Solution


The two original functions are ln(x) and ex 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 ex I get 1, and e. I don't really know how it helps but it's something I suppose.
 
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Your argument appears sound. Are you looking for a more algebraic justification? It shouldn't be too hard to turn your argument into algebra.
 
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.
 

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