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

Horrendous integration.

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

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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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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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