To find the coordinates of the centroid of the region bounded by the curves $y = x^2$ and $y = 1$, we need to follow these steps:

1. Find the Points of Intersection:

First, we determine where the curves intersect by setting $y = x^2$ equal to $y = 1$:

$x^2 = 1$

$x = \pm 1$

So, the curves intersect at the points $(-1, 1)$ and $(1, 1)$.

2. Set Up the Integral for the Centroid:

The centroid $(\bar{x}, \bar{y})$ of a region can be found using the formulas:

$\bar{x} = \frac{1}{A} \int_{a}^{b} x \left[ f(x) - g(x) \right] dx$

$\bar{y} = \frac{1}{A} \int_{a}^{b} \frac{1}{2} \left[ f(x)^2 - g(x)^2 \right] dx$

where:

• $f(x) = 1$ (the upper curve),
• $g(x) = x^2$ (the lower curve),
• $a = -1$, $b = 1$ (the limits of integration),
• $A$ is the area of the region.

3. Calculate the Area $A$:

The area $A$ of the region is given by:

$A = \int_{-1}^{1} \left(1 - x^2\right) dx$

Calculating this integral:

$A = \left[ x - \frac{x^3}{3} \right]_{-1}^{1}$

$A = \left(1 - \frac{1}{3}\right) - \left(-1 + \frac{1}{3}\right)$

$A = \left(\frac{2}{3} + \frac{2}{3}\right) = \frac{4}{3}$

4. Calculate $\bar{x}$:

Now, we calculate $\bar{x}$:

$\bar{x} = \frac{1}{A} \int_{-1}^{1} x \left(1 - x^2\right) dx$

$\bar{x} = \frac{3}{4} \int_{-1}^{1} \left(x - x^3\right) dx$

Since $\int_{-1}^{1} x dx = 0$ (because it is an odd function over a symmetric interval), we only need to compute:

$\bar{x} = \frac{3}{4} \left( \int_{-1}^{1} x dx - \int_{-1}^{1} x^3 dx \right)$

Both integrals for odd functions over symmetric intervals result in 0, so:

$\bar{x} = 0$

5. Calculate $\bar{y}$:

Now, we calculate $\bar{y}$:

$\bar{y} = \frac{1}{A} \int_{-1}^{1} \frac{1}{2} \left[1^2 - (x^2)^2\right] dx$

$\bar{y} = \frac{3}{4} \cdot \frac{1}{2} \int_{-1}^{1} \left(1 - x^4\right) dx$

$\bar{y} = \frac{3}{8} \left[ x - \frac{x^5}{5} \right]_{-1}^{1}$

$\bar{y} = \frac{3}{8} \left(2 - \frac{2}{5}\right) = \frac{3}{8} \cdot \frac{8}{5} = \frac{3}{5}$

Final Answer:

The centroid $(\bar{x}, \bar{y})$ of the region is:

$\left(0, \frac{3}{5}\right)$

Would you like any further details or have any questions? Here are five related questions you might find interesting:

1. How do you find the centroid of a region bounded by more complex curves?
2. What are the physical interpretations of the centroid in different contexts?
3. How does the centroid change when the region is not symmetric?
4. Can you determine the centroid of a 3D solid with a similar approach?
5. How do you calculate the area of more complex regions using double integrals?

Tip: Symmetry can simplify centroid calculations significantly. If the region is symmetric about a line, the centroid will lie on that line.