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#

#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]}

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Answer 2

To find the arc length of the curve ( f(x) = xe^{-x} ) in the interval ( x ) in ([0, \ln 7]), we use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + \left( f'(x) \right)^2} , dx ]

Where ( f'(x) ) is the derivative of ( f(x) ).

Taking the derivative of ( f(x) ):

[ f'(x) = e^{-x} - xe^{-x} ]

Now, we can substitute ( f'(x) ) into the arc length formula and integrate:

[ L = \int_{0}^{\ln 7} \sqrt{1 + \left( e^{-x} - xe^{-x} \right)^2} , dx ]

After simplifying and integrating, the arc length is found to be approximately ( 5.856 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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