To find the centroid of the region bounded by the curves $y = \sin(x)$, $y = 0$, $x = 0$, and $x = \pi$, we need to calculate the coordinates of the centroid $(\bar{x}, \bar{y})$. The formulas for the centroid of a region bounded by a function $y = f(x)$ over the interval $[a, b]$ are:

$\bar{x} = \frac{1}{A} \int_{a}^{b} x f(x) \, dx$

$\bar{y} = \frac{1}{A} \int_{a}^{b} \frac{f(x)^2}{2} \, dx$

Where $A$ is the area of the region.

Step 1: Calculate the Area $A$

The area $A$ under the curve $y = \sin(x)$ from $x = 0$ to $x = \pi$ is:

$A = \int_{0}^{\pi} \sin(x) \, dx$

Step 2: Calculate $\bar{x}$

$\bar{x} = \frac{1}{A} \int_{0}^{\pi} x \sin(x) \, dx$

Step 3: Calculate $\bar{y}$

$\bar{y} = \frac{1}{A} \int_{0}^{\pi} \frac{\sin^2(x)}{2} \, dx$

I'll solve these integrals step by step.The centroid $(\bar{x}, \bar{y})$ of the region bounded by the curves $y = \sin(x)$, $y = 0$, $x = 0$, and $x = \pi$ is:

$\bar{x} = \frac{\pi}{2} \approx 1.57$ $\bar{y} = \frac{\pi}{8} \approx 0.39$

The area of the region is $A = 2$.

Do you want more details on any of the steps, or have any questions?

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Here are five related questions for further exploration:

1. How would the centroid change if the region were bounded by $y = \sin(x)$ and $y = \cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$?
2. What is the centroid of the region bounded by $y = x^2$ and $y = 0$ from $x = 0$ to $x = 2$?
3. How does the method change if the region is not symmetric?
4. What is the physical significance of the centroid in engineering or physics?
5. How can numerical methods be used to approximate the centroid for more complex regions?

Tip: Remember that centroids represent the "average" location of a shape's area. In physical contexts, this is the center of mass if the material density is uniform.