# What is the arc length of f(x)=-xsinx+xcos(x-pi/2)  on x in [0,(pi)/4]?

Mar 21, 2018

$\frac{\pi}{4}$

#### Explanation:

The arc length of $f \left(x\right)$, $x \in \left[a . b\right]$ is given by:
${S}_{x} = {\int}_{b}^{a} f \left(x\right) \sqrt{1 + f ' {\left(x\right)}^{2}} \mathrm{dx}$

$f \left(x\right) = - x \sin x + x \cos \left(x - \frac{\pi}{2}\right) = - x \sin x + x \sin x = 0$
$f ' \left(x\right) = 0$

Since we just have $y = 0$ we can just take the length of s straight line between $0 \to \frac{\pi}{4}$ which is $\frac{\pi}{4} - 0 = \frac{\pi}{4}$