4. The profile of the cables of a suspension bridge may be modeled by a parabola. The centr span of the Golden Gate Bridge is 1280 meters long and 152 meters high. The parabola $$ y $$ $$ 0.00037 x^{2} $$ gives a good fit to the shape of the cables, where $$ |x| \leq 640 $$, and $$ x $$ and $$ y $$ are measured in meters. Find the definite integral that gives the length of the cables that stretch between the tops of the two towers (set-up but do not solve).

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To find the length of the cables modeled by the parabola $$ y = 0.00037x^2 $$ between the two towers of the Golden Gate Bridge, we'll use the formula for the arc length of a curve defined by a function $$ y = f(x) $$.

**Arc Length Formula:**
$$
L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
$$

**Step-by-Step Setup:**

1. **Identify the Function and Its Derivative:**
   $$
   y = 0.00037x^2
   $$
   $$
   \frac{dy}{dx} = 2 \times 0.00037x = 0.00074x
   $$

2. **Square the Derivative:**
   $$
   \left( \frac{dy}{dx} \right)^2 = (0.00074x)^2 = 0.0005476x^2
   $$

3. **Set Up the Integral Limits:**
   The span is from $$ x = -640 $$ meters to $$ x = 640 $$ meters.

4. **Write the Definite Integral for Arc Length:**
   $$
   L = \int_{-640}^{640} \sqrt{1 + 0.0005476x^2} \, dx
   $$

**Final Integral Expression:**
$$
\boxed{ \int_{-640}^{640} \sqrt{1 + \left(0.00074\,x\,\right)^2} \; dx }
$$

This integral represents the total length of the suspension bridge cables between the two tower tops. To find the numerical value of the length, you would evaluate this definite integral.
The length of the cables can be found using the integral: $$ \int_{-640}^{640} \sqrt{1 + (0.00074x)^2} \, dx $$

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