Finding path length of curve

**km707** · October 21st 2009, 01:53 AM

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!

**simplependulum** · October 21st 2009, 02:06 AM

so the derivative is $\frac{3x^2}{a^2} - \frac{a^2}{12x^2}$

$1 + (y')^2 = 1 + \left(\frac{3x^2}{a^2}\right)^2 - \frac{ 6 }{12}+ \left(\frac{a^2}{12x^2}\right)^2 = \left(\frac{3x^2}{a^2}\right)^2 + \frac{ 1 }{2}+ \left(\frac{a^2}{12x^2}\right)^2 = \left[\frac{3x^2}{a^2} + \frac{a^2}{12x^2}\right]^2$

that means you should then evaluate the integral

$\int \frac{3x^2}{a^2} + \frac{a^2}{12x^2} ~dx$

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