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

1 Answer
Aug 24, 2016

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]}