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

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Answer 1 · 2016-05-12T21:48:28

approximately $1.42326$ $1.42326$

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

$s=int_a^bsqrt(1+(f'(x))^2)dx$

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

$s=int_0^1sqrt(1+(5x^4-4x^3+1)^2)dx$

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

$sapprox1.42326$

What is the arclength of f(x)=x^5-x^4+x f(x)=x5−x4+xf(x)=x^5-x^4+x in the interval [0,1][0,1][0,1]?