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).

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.