We can calculate this using arc length formula.
[tex]L= \int\limits^a_b { \sqrt[]{1+ (\frac{dy}{dx} } )^2} \, dx [/tex]
The first step is to find the derivative of the given function.
[tex] \frac{dy}{dx}=(2 \sqrt{x^3})' [/tex]
[tex]( \frac{dy}{dx} )^2=9x[/tex]
Now we plug this back into original integral.
[tex]L= \int\limits^a_b { \sqrt[]{1+ 9x} \, dx [/tex]
We solve this integral using substituition u=9x+1. This way we end up solving elementary integral.
A final solution is:
[tex]L=\dfrac{2\left(9x+1\right)^\frac{3}{2}}{27}\mid_{25}^{250}[/tex]
Finaly we get that arc lenght is:
[tex]L=7659.29[/tex]
