Find the length of the curve y^2=x^3 from the origin where the tangent line makes an angle of 45 degrees with the x-axis.

Answer 1

since y=x^(3/2)
y' = (3/2)x^(1/2)
y'=1 when x = 4/9 and y = 8/27

so, the arc length is

s = ∫[0,4/9) √(1+((3/2)x^(1/2))^2) dx
= 1/2 ∫[0,4/9) √(9x+4) dx
= 8/27 (2√2-1)

As always, check my math