# What is the arclength of f(x)=sqrt((x-1)(x+2)-3x on x in [1,3]?

$s = 1.04487$ if we compute from $x = \left(1 + \sqrt{3}\right)$ to $x = 3$

there is no such length from $x = 1$ to $x = 3$

#### Explanation:

observe the graph $f \left(x\right) = \sqrt{\left(x - 1\right) \left(x + 2\right) - 3 x}$
graph{sqrt((x-1)(x+2)-3x) [-5, 5, -2.5, 2.5]}

the possible length is from $x = \left(1 + \sqrt{3}\right)$ to $x = 3$ and not from $x = 1$ to $x = 3$

$y ' = \frac{x - 1}{\sqrt{{x}^{2} - 2 x - 2}}$

s=int_(1+sqrt(3))^3 sqrt(1+((x-1)/sqrt(x^2-2x-2))^2