Zion Wheeler

2022-06-24

What is the arc length of $f\left(x\right)=\frac{3x}{\sqrt{x-1}}$ on $x\in \left[2,6\right]$?

enfujahl

Expert

Writing
$f\left(x\right)=3x\left(x-1{\right)}^{-\frac{1}{2}}$
then by the product and the chain rule we get
${f}^{\prime }\left(x\right)=3\left(x-1{\right)}^{-\frac{1}{2}}+3x\ast \left(-\frac{1}{2}\right)\ast \left(x-1{\right)}^{-\frac{3}{2}}$
which simplifies to
${f}^{\prime }\left(x\right)=\frac{3\left(x-2\right)}{2\ast \left(x-1{\right)}^{\frac{3}{2}}}$
so we have to solve
${\int }_{2}^{6}\sqrt{1+\left(3\frac{x-2}{2\ast \left(x-1{\right)}^{\frac{3}{2}}}{\right)}^{2}}dx$
by a numerical method we get
$\approx 4.51$

Do you have a similar question?