# What is the arc length of the curve given by #r(t)= (9sqrt(2),e^(9t),e^(-9t))# on # t in [3,4]#?

A zeroth-order approximation gives

Arc length is given by:

Simplify:

Factor out the larger piece:

A zeroth-order approximation gives:

Hence:

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To find the arc length of the curve given by ( r(t) = (9\sqrt{2}, e^{9t}, e^{-9t}) ) on ( t ) in the interval ([3,4]), we use the formula for arc length of a curve defined by vector-valued functions:

[ s = \int_{a}^{b} |r'(t)| , dt ]

where ( r(t) ) is the vector-valued function representing the curve, and ( |r'(t)| ) is the magnitude of its derivative.

In this case, the derivative ( r'(t) ) is:

[ r'(t) = \left(0, 9e^{9t}, -9e^{-9t}\right) ]

The magnitude of ( r'(t) ) is:

[ |r'(t)| = \sqrt{0^2 + (9e^{9t})^2 + (-9e^{-9t})^2} = \sqrt{81e^{18t} + 81e^{-18t}} ]

The integral for arc length becomes:

[ s = \int_{3}^{4} \sqrt{81e^{18t} + 81e^{-18t}} , dt ]

You can evaluate this integral to find the arc length of the curve over the given interval.

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