Finding the length of the curve calculus

Find the length of the curve y=(x−1)^(3/2) from (1,0) to (2,1).

arc lenght of f(x) is integral sqrt(f'(x)^2 + 1) dx

so f'(x)= 3/2 (x-1)^1/2
f'(x)^2= 9/4 (x-1)

so

integral sqrt((9/4)(x-1) + 1) dx = integral sqrt(9/4x - 9/4+1)dx= integral sqrt(9/4x -5/4) dx
factor out 1/4
integral 1/2 sqrt(9x-5) dx
so 1/2 * 1/9 * 2/3(9x-5)^(3/2)=
1/27 (9x-5)^3/2 evaluate from x=1 to x=2
we get

1/27 [(9*2-5)^3/2- (9-5)^3/2]
= 1/27 [13^3/2-4^3/2]= 1.43971 units