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3.7 Length of a graph

Proposition.

Suppose that y=f⁢(x)y=f(x) for a≤x≤ba\leq x\leq b. Then the length of the graph is

L=∫ab1+(d⁢fd⁢x)2⁢d⁢x.L=\int_{a}^{b}\sqrt{1+\Bigl({{df}\over{dx}}\Bigr)^{2}}\,dx.

For example, consider the positive branch y=x32y=x^{\frac{3}{2}} of Neil’s curve. Then d⁢yd⁢x=32⁢x.\frac{dy}{dx}=\frac{3}{2}\sqrt{x}. Thus the arc length between 00 and (t,t3/2)(t,t^{3/2}) is

∫0t1+94⁢x⁢d⁢x=32⁢∫0tx+49⁢d⁢x=32⁢[23⁢(x+49)3/2]0t{\int_{0}^{t}\sqrt{1+\frac{9}{4}x}\,dx=\,{\frac{3}{2}\int_{0}^{t}\sqrt{x+\frac% {4}{9}}\,dx}\,{=\frac{3}{2}\left[\frac{2}{3}\left(x+\frac{4}{9}\right)^{{3}/{2% }}\right]_{0}^{t}}}
=(x+49)3/2-(49)3/2=(x+49)3/2-827.{=\left(x+\frac{4}{9}\right)^{3/2}-\left(\frac{4}{9}\right)^{3/2}}\,{=\left(x+% \frac{4}{9}\right)^{3/2}-\frac{8}{27}.}