Question

Find the arc length function for the curve y = 2x3/2 with starting point p0(25, 250).

Answer 1

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