Question

What is the arclength from x=0 to x=9 of this curve: y=1/3 (√x)(x−3)

I can get most of the way there, but I run into mechanical problems when I try to actually evaluate the integral

Answer 1

\[Arc length = \int\limits_{0}^{9}\sqrt{1+f'(x)^{2}}dx\]
\[f'(x) = \frac{x-3}{6\sqrt{x}}+\frac{\sqrt{x}}{3} = \frac{x-1}{2\sqrt{x}}\]
\[\rightarrow \int\limits_{0}^{9}\frac{x+1}{2\sqrt{x}}dx\]
\[= \frac{1}{2}\int\limits_{0}^{9}(x^{1/2}+x^{-1/2})dx\]
\[=\frac{1}{3}x^{3/2}+x^{1/2} from 0->9\]
\[=12\]