Answer
$$\left. a \right)L = \int_1^4 {\sqrt {1 + \frac{{256}}{{{x^6}}}} } dx,\,\,\,\,\left. b \right)\,\,L \approx 8.71$$
Work Step by Step
$$\eqalign{
& y = \frac{8}{{{x^2}}}{\text{ on the interval }}\left[ {1,4} \right] \cr
& {\text{Definition of Arc Length for }}y = f\left( x \right): \cr
& {\text{Let }}f{\text{ have a continuous first derivative on the interval }}\left[ {a,b} \right]{\text{ }} \cr
& {\text{The length of the curve }} \cr
& {\text{from }}\left( {a,f\left( a \right)} \right){\text{ to }}\left( {b,f\left( b \right)} \right){\text{ is }}L = \int_a^b {\sqrt {1 + f'{{\left( x \right)}^2}} } dx \cr
& {\text{Notice that }}y = f\left( x \right) = \ln x{\text{ and }} \cr
& \left[ {1,4} \right] \to a = 1{\text{ and }}b = 4.{\text{ Then}} \cr
& \cr
& \left. a \right)\,\,f'\left( x \right) = \frac{d}{{dx}}\left[ {8{x^{ - 2}}} \right] \cr
& f'\left( x \right) = - 16{x^{ - 3}} \cr
& f'\left( x \right) = - \frac{{16}}{{{x^3}}} \cr
& {\text{Using the arc length formula}}{\text{, we have}} \cr
& L = \int_1^4 {\sqrt {1 + {{\left( { - \frac{{16}}{{{x^3}}}} \right)}^2}} } dx \cr
& L = \int_1^4 {\sqrt {1 + \frac{{256}}{{{x^6}}}} } dx \cr
& \cr
& \left. b \right){\text{ Use technology to evaluate or approximate the integral}} \cr
& L \approx 1.08 \cr} $$