# What is the arclength of f(x)=x^5-x^4+x  in the interval [0,1]?

May 12, 2016

approximately $1.42326$

#### Explanation:

The arc length of the function $f \left(x\right)$ on the interval $\left[a , b\right]$ can be found through:

$s = {\int}_{a}^{b} \sqrt{1 + {\left(f ' \left(x\right)\right)}^{2}} \mathrm{dx}$

So, here, we see that since $f \left(x\right) = {x}^{5} - {x}^{4} + x$, we know that $f ' \left(x\right) = 5 {x}^{4} - 4 {x}^{3} + 1$. Thus the arc length is equal to

$s = {\int}_{0}^{1} \sqrt{1 + {\left(5 {x}^{4} - 4 {x}^{3} + 1\right)}^{2}} \mathrm{dx}$

This can't be integrated by hand, so stick it into a calculator to see that

$s \approx 1.42326$