Find the coordinates of the centroid of the region bounded by

y equals 25 x squaredy=25x2

and

y equals 2.y=2.

The region is covered by a​ thin, flat plate.

To find the coordinates of the centroid of the region bounded by the curves \( y = 25x^2 \) and \( y = 2 \), we need to determine the points where these two curves intersect and then calculate the centroid using integrals.

### Step 1: Find Points of Intersection
Set \( y = 25x^2 \) equal to \( y = 2 \) to find the \( x \)-values of the intersection points.

\[
25x^2 = 2
\]
\[
x^2 = \frac{2}{25}
\]
\[
x = \pm \sqrt{\frac{2}{25}} = \pm \frac{\sqrt{2}}{5}
\]

Thus, the region is bounded horizontally from \( x = -\frac{\sqrt{2}}{5} \) to \( x = \frac{\sqrt{2}}{5} \).

### Step 2: Set Up the Integrals for the Centroid
The centroid \((\bar{x}, \bar{y})\) of a region can be found using the following formulas:
\[
\bar{x} = \frac{1}{A} \int_{a}^{b} x \, [f(x) - g(x)] \, dx
\]
\[
\bar{y} = \frac{1}{2A} \int_{a}^{b} [f(x)^2 - g(x)^2] \, dx
\]
where \( A \) is the area of the region, \( f(x) = 2 \), \( g(x) = 25x^2 \), and the limits \( a = -\frac{\sqrt{2}}{5} \) and \( b = \frac{\sqrt{2}}{5} \).

#### Area of the Region \( A \)
\[
A = \int_{a}^{b} [f(x) - g(x)] \, dx = \int_{-\frac{\sqrt{2}}{5}}^{\frac{\sqrt{2}}{5}} \left(2 - 25x^2\right) \, dx
\]

1. **Evaluate the integral** for \( A \).
2. **Plug values** into the formulas for \( \bar{x} \) and \( \bar{y} \).
3. **Simplify the expressions** to find the exact coordinates of the centroid.

Let me know if you need further detailed steps for these integrals or have any questions! 

### Related Questions:
1. How do we interpret centroids in physical applications?
2. Can centroids be outside of a bounded region?
3. How does symmetry affect the calculation of centroids?
4. What is the centroid of a circular region?
5. How does the centroid differ from the center of mass?

**Tip:** Symmetry often simplifies centroid calculations by reducing the need to calculate one of the coordinates.

Find the coordinates of the centroid of the region bounded by y = 25x^2 and y = 2. The region is covered by a thin, flat plate.

Learn how to calculate the centroid of the region bounded by the curves y = 25x^2 and y = 2 using integral calculus. This solution covers centroid formulas, area calculation through integration, and is ideal for college calculus students.

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