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

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Answer 1 · 2016-08-24T19:01:14

#approx 2.05#

#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]}