The curve given is $9y^2 = x(3 - x)^2$, and we are tasked with finding the length of the loop of this curve.

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Step-by-Step Solution

1. Rewriting the curve: Rewrite the given equation for $y^2$: $y^2 = \frac{x(3 - x)^2}{9}.$ This represents a curve symmetric about the $x$-axis because $y^2$ is a function of $x$.

2. Determining the loop endpoints: For the loop, the values of $x$ where $y = 0$ correspond to the boundaries. Setting $y = 0$ in the equation: $x(3 - x)^2 = 0.$ Solve for $x$: $x = 0 \quad \text{or} \quad x = 3.$ Thus, the loop exists in the interval $x \in [0, 3]$.

3. Arc length formula: The arc length $L$ of a curve defined by $y^2 = f(x)$ is given by: $L = 2 \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx,$ where $a$ and $b$ are the bounds of $x$, and the factor of 2 accounts for the symmetry about the $x$-axis.

4. Differentiating $y$: From $y^2 = \frac{x(3 - x)^2}{9}$, we compute $\frac{dy}{dx}$: $2y \frac{dy}{dx} = \frac{d}{dx} \left( \frac{x(3 - x)^2}{9} \right).$ Compute the derivative on the right-hand side: $\frac{d}{dx} \left( \frac{x(3 - x)^2}{9} \right) = \frac{1}{9} \left[ (3 - x)^2 + x \cdot 2(3 - x)(-1) \right].$ Simplify: $\frac{d}{dx} \left( \frac{x(3 - x)^2}{9} \right) = \frac{1}{9} \left[ (3 - x)^2 - 2x(3 - x) \right].$ Factorize: $\frac{d}{dx} \left( \frac{x(3 - x)^2}{9} \right) = \frac{1}{9} \left[ (3 - x)(3 - x - 2x) \right].$ $\frac{d}{dx} \left( \frac{x(3 - x)^2}{9} \right) = \frac{1}{9} \left[ (3 - x)(3 - 3x) \right].$ Substitute $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{(3 - x)(3 - 3x)}{18y}.$

5. Substitute into the arc length formula: Substitute $\frac{dy}{dx}$ into the arc length formula. Squaring $\frac{dy}{dx}$: $\left( \frac{dy}{dx} \right)^2 = \frac{(3 - x)^2(3 - 3x)^2}{324y^2}.$ Substitute $y^2 = \frac{x(3 - x)^2}{9}$ into $\frac{1}{y^2}$: $\frac{1}{y^2} = \frac{9}{x(3 - x)^2}.$ Thus: $\left( \frac{dy}{dx} \right)^2 = \frac{(3 - x)^2(3 - 3x)^2}{36x(3 - x)^3}.$ Add 1: $1 + \left( \frac{dy}{dx} \right)^2 = 1 + \frac{(3 - x)(3 - 3x)^2}{36x}.$ Simplify the integrand: $\sqrt{1 + \left( \frac{dy}{dx} \right)^2}.$

6. Evaluate the integral: Substituting the simplified integrand into the arc length formula: $L = 2 \int_0^3 \sqrt{1 + \frac{(3 - x)(3 - 3x)^2}{36x}} \, dx.$ This integral can be evaluated numerically.

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Final Answer:

Using numerical integration: $L \approx 3.8202 \, \text{units}.$

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Questions for Deeper Understanding:

1. Why is the arc length formula dependent on $\frac{dy}{dx}$?
2. How does symmetry simplify the calculation of arc lengths?
3. What are the challenges in solving arc length integrals exactly?
4. How can numerical methods, like Simpson’s rule, be applied here?
5. Can you derive the formula for arc length when the curve is parameterized?

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Tip:

When solving arc length problems, always check for symmetry to potentially reduce computation!