By the arc length formula, the length L {\displaystyle L} of the curve defined by the graph of y = f ( x ) {\displaystyle y=f(x)} from x = a {\displaystyle x=a} to x = b {\displaystyle x=b} is given by
So for the curve y = f ( x ) = 2 x 3 / 2 {\displaystyle y=f(x)=2x^{3/2}} from x = 0 {\displaystyle x=0} to x = 1 3 , {\displaystyle x={\frac {1}{3}},} we have
Using the substitution u = 1 + 9 x , d u = 9 d x , {\displaystyle u=1+9x,du=9\,dx,} the length can be computed as
of the curve defined by the graph of 
from 
to 
is given by
![{\displaystyle L=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,dx.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/01f9c77e776ed09477e79963fa5383c0d3cd7579)
from 
to 
we have
![{\displaystyle {\begin{aligned}L&=\int _{0}^{\frac {1}{3}}{\sqrt {1+[(2x^{\frac {3}{2}})']^{2}}}\,dx\\&=\int _{0}^{\frac {1}{3}}{\sqrt {1+(3x^{\frac {1}{2}})^{2}}}\,dx\\&=\int _{0}^{\frac {1}{3}}{\sqrt {1+9x}}\,dx.\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/2ba143964d5beedaf5d6f22aa7b92b06408141c3)
the length can be computed as
