# What is the arc length of f(x)= 1/sqrt(x-1)  on x in [2,4] ?

Mar 4, 2016

$L \approx 2.05411$

#### Explanation:

$f ' \left(x\right) = - \frac{1}{2 \sqrt{{\left(x - 1\right)}^{3}}}$

The length is given by

$L = {\int}_{2}^{4} \sqrt{1 + f ' {\left(x\right)}^{2}} \text{d} x$

$= {\int}_{2}^{4} \sqrt{1 + {\left(- \frac{1}{2 \sqrt{{\left(x - 1\right)}^{3}}}\right)}^{2}} \text{d} x$

$= {\int}_{2}^{4} \sqrt{1 + \frac{1}{4 {\left(x - 1\right)}^{3}}} \text{d} x$

$\approx 2.05411$