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

Arc length

The integral is complicated and requires the use of calculator or Simpson's Rule formula

God bless....I hope the explanation is useful.

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The arc length of the curve ( f(t) = (t - \sqrt{t^2 + 2}, t + te^{t - 2}) ) on the interval ([-1, 1]) can be calculated using the following formula:

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

Where ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ) are the derivatives of ( x ) and ( y ) with respect to ( t ), respectively.

So, first, compute ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ), then substitute them into the formula and integrate over the given interval ([-1, 1]) to find the arc length.

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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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