Question

What is the arc length of `f(x)= xsqrt(x^3-x+2)` on `x in [1,2]`?

Answer 1

`approx 4.3712`

Explanation:

We are using the formual
`\int_a^bsqrt(1+f'(x)^2)dx`

by the product and the chain rule we get

`f'(x)=sqrt(x^3-x+2)+x*1/2*(x^3-x+2)^(-1/2)*(3x^2-1)`

this is

`f'(x)=(2*(x^3-x+2)+3x^3-x)/(2sqrt(x^3-x+2))`

simplifying we get

`f'(x)=(5x^3-3x+4)/(2sqrt(x^3-x+2))`
and we have to integrate

`int_1^2sqrt(1+((5x^3-3x+4)/(2sqrt(x^3-x+2)))^2)dx`
By a numerical method we get `4.3712`