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Resulevelo y cual es la alternativa correcta

To solve this problem, we need to find the \( y \)-coordinate of the centroid (also known as the center of mass) for the area bounded by the curves \( y = x^2 - 20 \) and \( y = x \).

### Step 1: Find the Points of Intersection

We need to determine the points where the curves \( y = x^2 - 20 \) and \( y = x \) intersect. Set the equations equal to each other:

\[
x^2 - 20 = x
\]

Rearranging gives:

\[
x^2 - x - 20 = 0
\]

This is a quadratic equation. Solving for \( x \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1 \), \( b = -1 \), and \( c = -20 \). Substituting these values in:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1}
\]
\[
x = \frac{1 \pm \sqrt{1 + 80}}{2}
\]
\[
x = \frac{1 \pm \sqrt{81}}{2}
\]
\[
x = \frac{1 \pm 9}{2}
\]

So, the solutions are:

\[
x_1 = \frac{1 + 9}{2} = 5, \quad x_2 = \frac{1 - 9}{2} = -4
\]

The intersection points are at \( x = -4 \) and \( x = 5 \).

### Step 2: Determine the Area and the \( y \)-Coordinate of the Centroid

The \( y \)-coordinate of the centroid \( \bar{y} \) for the bounded area between two curves \( f(x) \) and \( g(x) \) from \( x = a \) to \( x = b \) is given by:

\[
\bar{y} = \frac{\int_{a}^{b} \left( \frac{f(x) + g(x)}{2} \right) \cdot (f(x) - g(x)) \, dx}{\int_{a}^{b} (f(x) - g(x)) \, dx}
\]

For our case, \( f(x) = x^2 - 20 \) and \( g(x) = x \), and the limits are from \( x = -4 \) to \( x = 5 \).

### Step 3: Set Up the Integrals

1. **Area Integral:**

\[
A = \int_{-4}^{5} ((x^2 - 20) - x) \, dx = \int_{-4}^{5} (x^2 - x - 20) \, dx
\]

2. **Centroid Integral:**

\[
\bar{y} = \frac{\int_{-4}^{5} \left( \frac{(x^2 - 20) + x}{2} \right) \cdot ((x^2 - 20) - x) \, dx}{A}
\]

Let's compute these integrals.The results of the integrals are:

1. **Area (A):** \(-121.5\) units (negative due to the direction, but we take the absolute value when considering the area).
2. **\( y \)-coordinate of the centroid ( \( \bar{y} \) ):** \(-7.6\) units.

### Conclusion

The \( y \)-coordinate of the centroid for the area shown is \(-7.6\) units. Thus, the correct answer is:

- **Alternative:** **\( -7.6 \, u \)**

Would you like more details or have any questions?

### Further Related Questions:

1. How do we calculate the centroid for more complex shapes?
2. What are the steps to compute the centroid for a 3D region?
3. Can you explain how to use the centroid formula for composite areas?
4. What happens if the curves intersect in more than two points?
5. How would the procedure change if we had different limits?

### Tip:

When solving centroid problems for bounded areas between curves, always verify the intersection points and ensure the curves are properly defined to avoid mistakes in integral limits and functions.

Determine la coordenada y del centroide del área mostrada

This problem involves finding the y-coordinate of the centroid for the area between two curves: y = x^2 - 20 and y = x. It covers curve intersection points, area calculation using integrals, and centroid formulas. Suitable for calculus students.

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