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

# 612.530 # (3dp)

We have:

The parametric arc-length is given by:

We can differentiate the parameters:

Then the arc-length is given by:

This integral dos not have a trivial anti-derivative, and so is evacuated using numerical methods to give:

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The arc length of the curve defined by ( f(t) = (3te^t, t - e^t) ) over the interval ( t ) in ([2, 4]) is given by the formula:

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

Substituting the given function ( f(t) = (3te^t, t - e^t) ), we find ( \frac{dx}{dt} = 3e^t + 3te^t ) and ( \frac{dy}{dt} = 1 - e^t ).

Plugging these into the arc length formula and integrating over the interval ([2, 4]) will give us 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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