The centroid of the region bounded by the curves \( y = \sin(x) \), \( y = 0 \), \( x = 0 \), and \( x = \pi \) is:

\[
\left( \bar{x}, \bar{y} \right) = \left( \frac{\pi}{2}, \frac{\pi}{8} \right) \approx (1.57, 0.39)
\]

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Here are five related questions for further exploration:

1. How do we find the centroid of a region bounded by polar curves?
2. What is the centroid of the region under \( y = e^x \) from \( x = 0 \) to \( x = 1 \)?
3. How does the shape of a curve affect the position of the centroid?
4. What happens to the centroid if the curve is translated?
5. Can the centroid be outside the region for some shapes?

**Tip:** The centroid can often be intuitively thought of as the "balance point" of the region's shape.

Discover how to find the centroid of the region bounded by the curves y = sin(x), y = 0, x = 0, and x = pi with a detailed step-by-step solution. This problem involves integral calculus and the concept of centroids in continuous regions. Suitable for advanced high school students and college-level calculus enthusiasts.

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