Question

What is the arc length of the curve given by `f(x)=xe^(-x)` in the interval `x in [0,ln7]`?

Answer 1

`approx 2.05`

Explanation:

`s = int dot s \ dt`

`= int_a^b sqrt(vec v * vec v) \ dt`

In Cartesian:
`vec r = ((x), (x e^(-x)))`

`vec v = d/dt ((x), (xe^(-x))) = ((dot x), ( dot x e^(-x) - dot x x e^(-x)))`

`= dot x ((1), ( e^(-x)(1- x)))`

`implies s = int_a^b sqrt(1 + e^(-2x) (1 - x )^2) \ dx/dt\ dt`

`implies int_0^(ln 7) sqrt(1 + e^(-2x) (1 - x )^2) \ dx`

Horrendous integration.

Computer says `approx 2.05`

graph{x e^(-x) [-1.25, 3.75, -0.73, 1.77]}