Question

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

Answer 1

`L = int_2^3 sqrt(1+9/4(x(x-1)(2x-1)^2)) dx`
Integrating we find (use Wolfram Alpha there is no closed integral for this integral):
L = 11.9145

Explanation:

We are going to use the Arc Length formula for L:
`L = int_a^bds`
where
`ds = sqrt(1+(dy/dx)^2) dx`
`L = int_a^b sqrt(1+(dy/dx)^2) dx`
so let's differentiate `y= f(x) = (x^2 - x)^(3/2)`
`dy/dx = f'(x) = 3/2 sqrt(x(x-1)) *(2x-1)`
Now find:
`(dy/dx)^2 = (f'(x))^2 = 9/4(x(x-1)(2x-1)^2)`
`L = int_2^3 sqrt(1+9/4(x(x-1)(2x-1)^2)) dx`
Integrating we find (use Wolfram Alpha there is no closed integral for this integral):
L = 11.9145