To find the arc length function for the curve defined by y = 2x^(3/2), we first need to use the arc length formula. The arc length L from point a to b for a curve y = f(x) is given by:
L = ∫(a to b) √(1 + (dy/dx)²) dx
First, we need to compute the derivative dy/dx. For our curve:
y = 2x^(3/2)
Taking the derivative:
 >dy/dx = 3x^(1/2)
Now we compute (dy/dx)²:
 >(dy/dx)² = (3x^(1/2))² = 9x
Next, we substitute (dy/dx)² into the arc length formula:
L = ∫(a to b) √(1 + 9x) dx
Now we set our limits of integration. Since we start from point P(0, 36) and want to find the arc length up to point (4, 32), our limits are from 0 to 4:
L = ∫(0 to 4) √(1 + 9x) dx
This integral can be solved by substituting u = 1 + 9x, which simplifies the integral. Differentiating gives us:
 >du = 9dx ⇒ dx = du/9
Now changing the limits accordingly: when x = 0, u = 1; when x = 4, u = 37.
Substituting into the integral, we get:
L = ∫(1 to 37) (1/9)√(u) du
Now integrating:
L = (1/9) * (2/3)u^(3/2) | (1 to 37) = (2/27)(37^(3/2) – 1^(3/2))
Calculating further:
 >L = (2/27)(37√37 – 1)
This gives us the arc length from the starting point P(0, 36) to the point (4, 32) on the curve. This expression represents the arc length function for the given curve.