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The problem in the image involves calculating the center of mass of a planar region enclosed by two curves:

The given curves are:
1. \( y = 0 \) (the x-axis)
2. \( y = 4 - x^2 - 2x \)

We need to compute the coordinates of the center of mass (\( \bar{x} \), \( \bar{y} \)) using the given formulas:

1. \( \bar{x} = \frac{1}{A} \int_a^b x(f(x) - g(x)) \, dx \)
2. \( \bar{y} = \frac{1}{2A} \int_a^b (f(x)^2 - g(x)^2) \, dx \)

Where:
- \( f(x) = 4 - x^2 - 2x \) is the upper curve.
- \( g(x) = 0 \) is the lower curve (the x-axis).
- \( A \) is the area of the region.

Let me proceed by calculating each step starting with the limits of integration \(a\) and \(b\), which are determined by solving the intersection points of the curves \( y = 4 - x^2 - 2x \) and \( y = 0 \). 

Let's solve this.The intersection points between the curves \( y = 4 - x^2 - 2x \) and \( y = 0 \) are:

\[
x_1 = -1 + \sqrt{5}, \quad x_2 = -1 - \sqrt{5}
\]

These will serve as the limits of integration for calculating the area and the coordinates of the center of mass. Now, let's compute the area \( A \) of the region using the following formula:

\[
A = \int_{x_2}^{x_1} (f(x) - g(x)) \, dx = \int_{x_2}^{x_1} (4 - x^2 - 2x) \, dx
\]

I'll compute the area first.The area \( A \) of the region is:

\[
A = \frac{20 \sqrt{5}}{3}
\]

Now, let's compute the coordinates of the center of mass. First, we will calculate \( \bar{x} \) using the formula:

\[
\bar{x} = \frac{1}{A} \int_{x_2}^{x_1} x(f(x) - g(x)) \, dx = \frac{1}{A} \int_{x_2}^{x_1} x(4 - x^2 - 2x) \, dx
\]

I will proceed with this calculation.The \( \bar{x} \)-coordinate of the center of mass is:

\[
\bar{x} = -1
\]

Next, let's calculate \( \bar{y} \) using the formula:

\[
\bar{y} = \frac{1}{2A} \int_{x_2}^{x_1} \left( f(x)^2 - g(x)^2 \right) \, dx = \frac{1}{2A} \int_{x_2}^{x_1} \left( (4 - x^2 - 2x)^2 - 0^2 \right) \, dx
\]

I will proceed with the computation of \( \bar{y} \).It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through the steps to calculate the \( \bar{y} \)-coordinate manually. Let me know how you'd like to proceed!

Would you like more details or clarifications? Here are five related questions that can help deepen your understanding:
1. How is the area of a region between curves calculated in general?
2. Why does the formula for the center of mass include the integrals of \( x \) and \( y^2 \)?
3. What does the center of mass represent in physical terms for this region?
4. How would the process change if the curves were not symmetric?
5. What role do the intersection points play in defining the limits of integration?

**Tip:** Understanding the geometric meaning of the center of mass helps in visualizing the balance point of a region.

Calculate the center of mass of a planar region enclosed by the curves y = 0 and y = 4 - x^2 - 2x.

Learn how to calculate the center of mass of a region enclosed by the curves y = 0 and y = 4 - x^2 - 2x. This problem involves key calculus concepts such as integration and the computation of the area between curves.

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