doturitip9

2022-07-03

What is the arc length of $f\left(x\right)=-x\mathrm{sin}x+x\mathrm{cos}\left(x-\frac{\pi }{2}\right)$ on $x\in \left[0,\frac{\pi }{4}\right]$?

Yair Boyle

Expert

Explanation:
The arc length of $f\left(x\right),x\in \left[a,b\right]$ is given by:
${S}_{x}={\int }_{b}^{a}f\left(x\right)\sqrt{1+{f}^{\prime }\left(x{\right)}^{2}}dx$
$f\left(x\right)=-x\mathrm{sin}x+x\mathrm{cos}\left(x-\frac{\pi }{2}\right)=-x\mathrm{sin}x+x\mathrm{sin}x=0$
f'(x)=0
Since we just have y=0 we can just take the length of s straight line between
$0\to \frac{\pi }{4}whichis\frac{\pi }{4}-0=\frac{\pi }{4}$

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