# What is the arclength of #f(t) = (t-sqrt(t-1),t^2/(t^2-1))# on #t in [2,3]#?

Result obtained using numerical integration, can't find closed form antiderivative:

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To find the arc length of the curve ( f(t) = (t - \sqrt{t - 1}, \frac{t^2}{t^2 - 1}) ) on the interval ( t \in [2,3] ), 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 ]

For ( f(t) ), we first find ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ):

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

Now, we substitute these derivatives into the arc length formula and integrate over the interval ( t \in [2,3] ):

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

This integral may not have a simple closed-form solution and may need to be evaluated numerically.

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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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