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

In summary, the two given functions, ln(x) and ex, are symmetrical about the line y = x. This means that their arc lengths over a certain region will be the same. When the bounds 1 and e are plugged into ln(x), the result is 0 and 1, respectively. When the bounds 0 and 1 are plugged into ex, the result is 1 and e, respectively. This provides an algebraic justification for why the integrals of the two functions over the given intervals are equal.
  • #1
icesalmon
270
13

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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  • #2
Your argument appears sound. Are you looking for a more algebraic justification? It shouldn't be too hard to turn your argument into algebra.
 
  • #3
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.
 

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

1. What is the symmetric arc length of ln(x)?

The symmetric arc length of ln(x) is equal to the integral of the natural logarithm function from 1 to e, which is approximately 1.783.

2. How is the symmetric arc length of ln(x) different from the arc length of ln(x)?

The symmetric arc length of ln(x) takes into account the symmetry of the natural logarithm function, which means that the arc length from 1 to e is the same as the arc length from e to infinity. This is not the case for the regular arc length, which only measures the distance from 1 to a specific value of x.

3. What is the relationship between the symmetric arc length of ln(x) and the arc length of e^x?

The symmetric arc length of ln(x) and the arc length of e^x are equal because ln(x) and e^x are inverse functions of each other. This means that the area under the curve of ln(x) is the same as the area under the curve of e^x, resulting in equal arc lengths.

4. How is the symmetric arc length of ln(x) useful in mathematics?

The symmetric arc length of ln(x) is useful in calculating the arc length of other inverse functions, such as the logarithmic function with a different base. It also has applications in physics and engineering, where it is used to calculate the work done by a force in a conservative system.

5. Can the symmetric arc length of ln(x) be calculated using any other method?

Yes, the symmetric arc length of ln(x) can also be calculated using the substitution method or the integration by parts method. However, the formula for the symmetric arc length provides a more direct and simplified approach to the calculation.

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