Final answer:
To find the exact length of the curve, use the arc length formula and integrate the expression.
Step-by-step explanation:
To find the exact length of the curve given by the equation x = (1/3) * √y * (y - 3) where 9 ≤ y ≤ 25, we can use the arc length formula for a curve y = f(x) between two points a and b:
L = ∫ab √(1 + (dy/dx)^2) dx
First, we need to express the equation in terms of x. Since x = (1/3) * √y * (y - 3), we can rewrite it as y = (3x + 9)/(x+1). Now we can differentiate y with respect to x to find the value of (dy/dx).
Next, we substitute the values of a = 9 and b = 25 into the arc length formula and evaluate the integral to find the exact length of the curve.