# What is the arc length of #f(t)=(t^2-4t,5-t) # over #t in [3,4] #?

Arc length is given by:

This is a known integral:

Reverse the substitution:

Insert the limits of integration:

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To find the arc length of the curve represented by the parametric equations ( f(t) = (t^2 - 4t, 5 - t) ) over the interval ( t ) in ([3,4]), we use the formula for arc length:

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

We first find the derivatives ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ):

[ \frac{dx}{dt} = \frac{d}{dt}(t^2 - 4t) = 2t - 4 ] [ \frac{dy}{dt} = \frac{d}{dt}(5 - t) = -1 ]

Substituting these derivatives into the arc length formula:

[ L = \int_{3}^{4} \sqrt{(2t - 4)^2 + (-1)^2} dt ]

[ L = \int_{3}^{4} \sqrt{4t^2 - 16t + 17} dt ]

[ L = \int_{3}^{4} \sqrt{4(t^2 - 4t + \frac{17}{4})} dt ]

[ L = \int_{3}^{4} 2\sqrt{t^2 - 4t + \frac{17}{4}} dt ]

[ L = 2\int_{3}^{4} \sqrt{\left(t - 2\right)^2 + \frac{1}{4}} dt ]

[ L = 2\left[\frac{1}{2} \ln\left| 2t - 4 + 2\sqrt{t^2 - 4t + \frac{17}{4}}\right|\right]_{3}^{4} ]

[ L = \ln\left| 2\sqrt{t^2 - 4t + \frac{17}{4}} + 2t - 4\right| \Bigg|_{3}^{4} ]

[ L = \ln\left| 2\sqrt{4 - 4(4) + \frac{17}{4}} + 2(4) - 4\right| - \ln\left| 2\sqrt{9 - 4(3) + \frac{17}{4}} + 2(3) - 4\right| ]

[ L = \ln\left| 2\sqrt{1} + 4\right| - \ln\left| 2\sqrt{\frac{1}{4}} + 2\right| ]

[ L = \ln\left| 2 + 4\right| - \ln\left| 2 + 2\right| ]

[ L = \ln(6) - \ln(4) ]

[ L = \ln\left(\frac{3}{2}\right) ]

Therefore, the arc length of the curve over the interval ( t ) in ([3,4]) is ( \ln\left(\frac{3}{2}\right) ).

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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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