# What is the arc length of the curve given by #r(t)= (1/t,1,t)# on # t in [0, 1]#?

Not measurable, The curve has a nonintegrable singularity at

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To find the arc length of the curve given by ( r(t) = \left(\frac{1}{t}, 1, t\right) ) on ( t ) in ([0, 1]), we use the formula for arc length:

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

For ( r(t) = \left(\frac{1}{t}, 1, t\right) ), the derivatives are:

[ \frac{dx}{dt} = -\frac{1}{t^2} ] [ \frac{dy}{dt} = 0 ] [ \frac{dz}{dt} = 1 ]

Plugging these into the arc length formula:

[ s = \int_{0}^{1} \sqrt{\left(-\frac{1}{t^2}\right)^2 + (0)^2 + (1)^2} dt ]

[ s = \int_{0}^{1} \sqrt{\frac{1}{t^4} + 1} dt ]

[ s = \int_{0}^{1} \sqrt{\frac{1 + t^4}{t^4}} dt ]

[ s = \int_{0}^{1} \sqrt{\frac{1}{t^4} + 1} dt ]

[ s = \int_{0}^{1} \frac{\sqrt{1 + t^4}}{t^2} dt ]

This integral may not have a simple closed-form solution, so it would typically be evaluated numerically.

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