Question

What is the arclength of `f(x)=x/e^(3x)` on `x in [1,2]`?

Answer 1

Approximately 1.

Explanation:

The arc length is calculated from the following integral:

`L=int_a^bsqrt(1+((df(x))/dx)^2)dx`

which leads us to

`L=int_1^2sqrt(1+[(1-3x)e^(-3x)]^2)dx`

I haven't found an analytic expression for the integral. However, when we plot the function `f(x)` it becomes clear that the function has decayed to very small values.

What is deceiving about this graph is that arc-length is calculated in normal Cartesian space, which implies that the scale of `x` and `y` should be equal as in the following version:

This implies that the actual arc length between 1 and 2 will be very close to the distance on the axis. Doing the integration numerically, one gets

`L~=1.0013...`

which is very close to 1 as expected. A possible refinement would be to approximate the arc length with a straight line between `f(1)` and `f(2)` giving the two points

`(1, 0.050) and (2, 0.005)`

`L~=sqrt(1+(0.050-0.005)^2)=1.0010...`