**Let n mayor o igual que 1 be constant, and consider the region plane bounded by f paréntesis izquierdo x paréntesis derecho igual 6 espacio x elevado a n, the x-axis, and x igual 1. The value of centroid envoltorio arriba x when n flecha derecha infinito is: Respuesta The valua of centroid envoltorio arriba y when n flecha derecha infinito is: Respuesta**

We are asked to compute the centroid of a region bounded by the function \( f(x) = 6x^n \), the x-axis, and the vertical line \( x = 1 \), and then analyze the behavior as \( n \to \infty \).

### Step 1: Recall the centroid formulas

The coordinates of the centroid \( (x_{\text{centroid}}, y_{\text{centroid}}) \) of a region are given by:

\[
x_{\text{centroid}} = \frac{1}{A} \int_a^b x f(x) \, dx
\]
\[
y_{\text{centroid}} = \frac{1}{2A} \int_a^b [f(x)]^2 \, dx
\]
where \( A \) is the area of the region, and the bounds are from \( x = 0 \) to \( x = 1 \).

### Step 2: Calculate the area \( A \)

The area \( A \) under the curve is:

\[
A = \int_0^1 6x^n \, dx
\]

Solving this integral:

\[
A = 6 \int_0^1 x^n \, dx = 6 \left[ \frac{x^{n+1}}{n+1} \right]_0^1 = 6 \cdot \frac{1}{n+1} = \frac{6}{n+1}
\]

### Step 3: Compute \( x_{\text{centroid}} \)

Now, we calculate \( x_{\text{centroid}} \):

\[
x_{\text{centroid}} = \frac{1}{A} \int_0^1 x(6x^n) \, dx = \frac{1}{A} \int_0^1 6x^{n+1} \, dx
\]

Solving the integral:

\[
x_{\text{centroid}} = \frac{1}{\frac{6}{n+1}} \cdot 6 \left[ \frac{x^{n+2}}{n+2} \right]_0^1 = \frac{n+1}{1} \cdot \frac{6}{n+2} = \frac{n+1}{n+2}
\]

As \( n \to \infty \), \( x_{\text{centroid}} \) approaches:

\[
x_{\text{centroid}} = \lim_{n \to \infty} \frac{n+1}{n+2} = 1
\]

### Step 4: Compute \( y_{\text{centroid}} \)

Next, we compute \( y_{\text{centroid}} \):

\[
y_{\text{centroid}} = \frac{1}{2A} \int_0^1 [6x^n]^2 \, dx = \frac{1}{2A} \int_0^1 36x^{2n} \, dx
\]

Solving the integral:

\[
y_{\text{centroid}} = \frac{1}{2A} \cdot 36 \left[ \frac{x^{2n+1}}{2n+1} \right]_0^1 = \frac{1}{2A} \cdot 36 \cdot \frac{1}{2n+1}
\]
\[
y_{\text{centroid}} = \frac{1}{2 \cdot \frac{6}{n+1}} \cdot \frac{36}{2n+1} = \frac{(n+1) \cdot 36}{12(2n+1)} = \frac{3(n+1)}{2n+1}
\]

As \( n \to \infty \), \( y_{\text{centroid}} \) approaches:

\[
y_{\text{centroid}} = \lim_{n \to \infty} \frac{3(n+1)}{2n+1} = \frac{3}{2}
\]

### Final Answers

- \( x_{\text{centroid}} \to 1 \) as \( n \to \infty \)
- \( y_{\text{centroid}} \to \frac{3}{2} \) as \( n \to \infty \)

Would you like more details on any part of this solution? Here are 5 related questions for further exploration:

1. How does the centroid change for different functions other than \( f(x) = 6x^n \)?
2. What is the physical meaning of a centroid in terms of balance?
3. How would the centroid change if the function were bounded by a different vertical line than \( x = 1 \)?
4. How can we visualize the centroid's movement as \( n \) increases?
5. Can this method be applied to non-polynomial functions?

**Tip:** The centroid (center of mass) of a shape can give insight into its balance point and stability.

Let n ≥ 1 be constant, and consider the region bounded by f(x) = 6x^n, the x-axis, and x = 1. The value of the centroid x as n → ∞ is: The value of the centroid y as n → ∞ is:

Explore how to compute the centroid of a region bounded by the curve f(x) = 6x^n, the x-axis, and the line x = 1. Learn how to apply integration and limit concepts to find the coordinates of the centroid as n approaches infinity. Suitable for college-level calculus students.

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