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We are tasked with finding the exact coordinates of the centroid of the region bounded by the curves \( y = x + 56 \) and \( y = x^2 \).

### Steps to Solve:

1. **Find the points of intersection**: 
   To determine the region bounded by the curves, we need to find the points where \( y = x + 56 \) and \( y = x^2 \) intersect. Set the equations equal to each other:
   \[
   x + 56 = x^2
   \]
   Rearranging this equation:
   \[
   x^2 - x - 56 = 0
   \]
   Solve this quadratic equation using the quadratic formula:
   \[
   x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-56)}}{2(1)}
   \]
   Simplifying:
   \[
   x = \frac{1 \pm \sqrt{1 + 224}}{2} = \frac{1 \pm \sqrt{225}}{2} = \frac{1 \pm 15}{2}
   \]
   So the solutions are:
   \[
   x = \frac{1 + 15}{2} = 8 \quad \text{and} \quad x = \frac{1 - 15}{2} = -7
   \]
   Therefore, the curves intersect at \( x = 8 \) and \( x = -7 \).

2. **Area of the region**: 
   The centroid \( (\bar{x}, \bar{y}) \) is given by:
   \[
   \bar{x} = \frac{1}{A} \int_{a}^{b} x \left( f(x) - g(x) \right) \, dx
   \]
   \[
   \bar{y} = \frac{1}{2A} \int_{a}^{b} \left( f(x)^2 - g(x)^2 \right) \, dx
   \]
   where \( f(x) = x + 56 \), \( g(x) = x^2 \), and \( A \) is the area of the region. 

   The area of the region is:
   \[
   A = \int_{-7}^{8} \left( (x + 56) - x^2 \right) \, dx
   \]
   Let's solve for the area \( A \).

3. **Find \( \bar{x} \) and \( \bar{y} \)**:
   Next, we'll compute the integrals for \( \bar{x} \) and \( \bar{y} \), based on the formulas mentioned above.

I will now compute these steps.

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Would you like more details on the individual integration steps, or should I continue with the complete solution?

Find the exact coordinates of the centroid of the region bounded by the curves y = x + 56 and y = x^2.

This problem involves calculating the exact coordinates of the centroid of the region bounded by the curves y = x + 56 and y = x^2 using integral calculus and the centroid formulas. Suitable for college-level students studying calculus.

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