# What is the arclength of f(x)=sqrt((x-1)(2x+2))-2x on x in [6,7]?

Jun 2, 2018

$1.15037$

#### Explanation:

We have
$f \left(x\right) = \sqrt{\left(x - 1\right) \cdot 2 \cdot \left(x + 1\right)} - 2 x$
$f \left(x\right) = \sqrt{2} \cdot \sqrt{{x}^{2} - 1} - 2 x$
So we get the integral

${\int}_{6}^{7} \sqrt{1 + {\left(\sqrt{2} \cdot \frac{x}{\sqrt{{x}^{2} - 1}} - 2\right)}^{2}} \mathrm{dx}$
With a numerical method we find
$\approx 1.1 .5037$