Question

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

Answer 1

approximately `1.42326`

Explanation:

The arc length of the function `f(x)` on the interval `[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`