Answer
$\dfrac{123}{32}$
Work Step by Step
The formula to calculate the arc length is as follows:
$L=\int_m^n \sqrt {1+[f'(x)]^2} dx$
Re-write the equation as follows: $[f'(x)]^2=\dfrac{(y-1)^2}{4y}$
This implies that $L=\int_1^2 [1+\dfrac{(4y^6-1)^2}{16y^6}] dy $
Separate the terms and integrate as follows: $L=\int_1^2 y^3+\dfrac{y^{-3}}{4} dy \\=[ \dfrac{y^4}{4}-\dfrac{1}{8y^2} ]_1^{2}=\dfrac{123}{32}$
