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What is the arc length of #f(x)=(1-x)e^(4-x) # on #x in [1,4] #?

Answer 1

Emily Ammerman

#int_1^4 sqrt(1 + ((x - 2)e^(4 -x))^2)dx = ≈12.4385#

Arc Length

#s = int_a^b sqrt(1 + (f'(x))^2)dx#

Use WolframAlpha to compute f'(x)

#f'(x) = (x - 2)e^(4 -x)#

The indefinite integral cannot be done using standard mathematical functions but I was able to make WolframAlpha evaluate the definite integral

#int_1^4 sqrt(1 + ((x - 2)e^(4 -x))^2)dx = ≈12.4385#

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Answer 2

Elijah Abram

To find the arc length of the function ( f(x) = (1 - x)e^{4 - x} ) on the interval ([1, 4]), we use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx ]

Where ( a = 1 ), ( b = 4 ), and ( \frac{dy}{dx} ) is the derivative of ( f(x) ).

Taking the derivative of ( f(x) ) gives:

[ \frac{dy}{dx} = -e^{4-x} - (1 - x)e^{4-x} ]

Then, we plug this derivative into the formula for arc length and integrate from ( x = 1 ) to ( x = 4 ):

[ L = \int_{1}^{4} \sqrt{1 + (-e^{4-x} - (1 - x)e^{4-x})^2} dx ]

Solving this integral will give us the arc length of the function on the specified interval.

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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.