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To find the length of the curve, we use the arc length formula for a curve given parametrically or as \( x = f(y) \) over a certain range of \( y \).

### Arc Length Formula:
For a function \( x = f(y) \), the length of the curve from \( y = a \) to \( y = b \) is given by:

\[
L = \int_a^b \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy
\]

### Given Function:
The given function is:
\[
x = \frac{1}{3} \sqrt{y(y - 3)}
\]
with \( 1 \leq y \leq 9 \).

### Step 1: Find \( \frac{dx}{dy} \)
We first need to find the derivative \( \frac{dx}{dy} \).

\[
x = \frac{1}{3} \sqrt{y(y - 3)} = \frac{1}{3} \sqrt{y^2 - 3y}
\]

Using the chain rule to differentiate with respect to \( y \):

\[
\frac{dx}{dy} = \frac{1}{3} \cdot \frac{1}{2} (y^2 - 3y)^{-\frac{1}{2}} \cdot (2y - 3)
\]

This simplifies to:

\[
\frac{dx}{dy} = \frac{2y - 3}{6 \sqrt{y^2 - 3y}}
\]

### Step 2: Set up the Arc Length Integral
Now, substitute \( \frac{dx}{dy} \) into the arc length formula:

\[
L = \int_1^9 \sqrt{1 + \left( \frac{2y - 3}{6 \sqrt{y^2 - 3y}} \right)^2} \, dy
\]

### Step 3: Simplify and Solve the Integral
This integral is quite complex and might require substitution or numerical methods to evaluate. We can proceed with the evaluation either symbolically (if simplifications can be made) or using a numerical integration approach.

Would you like the detailed solution to proceed further with solving this, or should I perform a numerical evaluation for the exact length?

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### Questions to Expand Understanding:
1. How does the chain rule apply when differentiating composite functions?
2. What are some common methods for simplifying complex integrals like this one?
3. Can the arc length formula be applied to curves given in parametric form?
4. Why is the derivative \( \frac{dx}{dy} \) crucial in the arc length formula?
5. Are there alternative methods for approximating arc length for non-symbolic functions?

### Tip:
When dealing with square roots in derivatives, check for possible simplifications before plugging them into more complex formulas like arc length to ease the integration process.

Find the length of the curve where x = (1/3) * sqrt(y(y-3)) for 1 ≤ y ≤ 9.

Learn how to find the arc length of the curve defined by x = (1/3) * sqrt(y(y-3)) over the interval 1 ≤ y ≤ 9. This problem involves the arc length formula and requires calculus techniques such as differentiation and applying the chain rule.

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