Hailie Blevins

2022-06-29

What is the arclength of $f\left(x\right)=\sqrt{\left(x-1\right)\left(2x+2\right)}-2x$ on $x\in \left[6,7\right]$?

tennispopj8

Expert

Explanation:
We have
$f\left(x\right)=\sqrt{\left(x-1\right)\ast 2\ast \left(x+1\right)}-2x$
$f\left(x\right)=\sqrt{2}\ast \sqrt{{x}^{2}-1}-2x$
So we get the integral
${\int }_{6}^{7}\sqrt{1+\left(\sqrt{2}\ast \frac{x}{\sqrt{{x}^{2}-1}}-2{\right)}^{2}}dx$
With a numerical method we find
$\approx 1.15037$

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