Find the path length of y= (x^3)/(a^2) + (a^2)/(12x)
I know I'm supposed to be integrating ds = dx (1+ (dy/dx)^2)^1/2
but i'm having trouble simplifying the expression inside the square root..
thanks in advance!
so the derivative is $\displaystyle \frac{3x^2}{a^2} - \frac{a^2}{12x^2}$
$\displaystyle 1 + (y')^2 = 1 + \left(\frac{3x^2}{a^2}\right)^2 - \frac{ 6 }{12}+ \left(\frac{a^2}{12x^2}\right)^2 = \left(\frac{3x^2}{a^2}\right)^2 + \frac{ 1 }{2}+ \left(\frac{a^2}{12x^2}\right)^2 = \left[\frac{3x^2}{a^2} + \frac{a^2}{12x^2}\right]^2$
that means you should then evaluate the integral
$\displaystyle \int \frac{3x^2}{a^2} + \frac{a^2}{12x^2} ~dx$