Answer
$309.3195$
Work Step by Step
The formula for the arc length is
$s=\int_{a}^b \sqrt{1 + (y')^2} dy$
Differentiate $y = \frac{x^5}{10}+\frac{1}{6x^3}$ with respect to x.
$y' = \frac{x^4}{2}- \frac{1}{2x^4}$
$=\frac{1}{2}(x^4 - x^{−4})$
Substitute the value of y' in $1 + (y')^2$
$1+(y')^2=1+[\frac{1}{2}(x^4 - x^{−4})]^2$
$=[\frac{1}{2}(x^4 + x^{−4})]^2$
Substitute the value of $1 + (y')^2$ in the formula for the arc length $s=\int_{a}^b \sqrt{1 + (y')^2} dy$
Solve for the distance s
$s=\int_{2}^5\sqrt{\frac{1}{2}(x^4 + x^{-4})}^{2} dx$
$=\frac{1}{2}\int_{2}^5 (x^4 + x^{-4})dx$
$=\frac{1}{2}[\frac{x^5}{5}-\frac{1}{3x^3}]_2^5\ $
$=309.3195$
