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

The arc length formula for a parametric equation is:

Now plug into the formula and solve:

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To find the arc length of the curve described by the parametric equations ( x(t) = e^{2t} - t ) and ( y(t) = t - \frac{t}{e^{t-1}} ) on the interval ( t ) in ([-1,1]), we use the formula:

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

Where:

- ( \frac{dx}{dt} ) represents the derivative of ( x(t) ) with respect to ( t )
- ( \frac{dy}{dt} ) represents the derivative of ( y(t) ) with respect to ( t )
- ( a ) and ( b ) are the lower and upper limits of the parameter ( t ), respectively.

By calculating the derivatives, we get:

[ \frac{dx}{dt} = 2e^{2t} - 1 ] [ \frac{dy}{dt} = 1 - \frac{1}{e^{t-1}} + \frac{t}{e^{t-1}} ]

Now, we substitute these derivatives into the arc length formula and integrate over the interval ([-1,1]):

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

This integral needs to be solved numerically, as it does not have a simple closed form solution. You can use numerical integration techniques such as the trapezoidal rule or Simpson's rule to approximate the value of the integral.

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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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