Problem 77 - Essential Calculus

Question

Let \(n \geq 1\) be constant, and consider the region bounded by \(f(x)=x^{n},\) the \(x\) -axis, and \(x=1\). Find the centroid of this region. As \(n \rightarrow \infty\), what does the region look like, and where is its centroid?

Step-by-Step Solution

Answer

The centroid of the region bounded by the curve \(f(x) = x^n\), the x-axis, and \(x = 1\) is \((\frac{n+1}{n+2},\frac{n+1}{2n+2})\). As \(n \rightarrow \infty\), the centroid is \((1,0.5)\), resembling a vertical line at \(x = 1\).

Step 1: Calculating the Area under f(x)

The area 'A' under the curve \( f(x) = x^n \) from 0 to 1 is given by: \( A = \int_{0}^{1} x^n dx \), computing this we get \( A = \frac{1}{n+1} \).

Step 2: Calculating the centroid coordinates

Next, calculate the x-coordinate of the centroid \(\bar{x}\) using the formula: \(\bar{x}= \frac{1}{A} \int_{0}^{1} x. f(x) dx \) which calculates to \(\bar{x}= \frac{1}{\frac{1}{n+1}} \int_{0}^{1} x^{n+1} dx = \frac{n+1}{n+2} \). Similarly, calculate the y-coordinate of the centroid \(\bar{y}\) using the formula: \( \bar{y}= \frac{1}{2A} \int_{0}^{1} (f(x))^2 dx \) which calculates to \(\bar{y} = \frac{1}{2\frac{1}{n+1}} \int_{0}^{1} x^{2n} dx = \frac{n+1}{2n+2} \).

Step 3: Describing the region and its centroid as \( n \rightarrow \infty \)

As \( n \rightarrow \infty \), the graph of the function \( x^n \) approaches the x-axis for \(0

Key Concepts

Definite Integrals

A definite integral helps us find the area under a curve between two points. It is a fundamental concept in calculus essential for various applications. In this problem, we need to calculate the area under the function \( f(x) = x^n \) from \( x = 0 \) to \( x = 1 \). Here's how it works:

• The integral \( \int_{0}^{1} x^n \, dx \) is computed, resulting in \( \frac{1}{n+1} \). This value represents the total area under the curve, which is crucial for further calculations.
• Definite integrals give you exact numerical values rather than a function, thus allowing you to determine quantities like area with precise bounds.

The role of the definite integral in this exercise provides the baseline for determining the region's centroid.

Centroid Coordinates

The centroid of a region acts similarly to a center of mass, balancing the shape perfectly. To find it, we need coordinates \( \bar{x} \) and \( \bar{y} \). Here’s how each is determined:

• X-coordinate \( \bar{x} \): We use the formula \( \bar{x} = \frac{1}{A} \int_{0}^{1} x \cdot f(x) \, dx \). With the previously found area, \( A = \frac{1}{n+1} \), it evaluates to \( \bar{x} = \frac{n+1}{n+2} \). This coordinate leans towards the higher \( x \) values due to the curve's stretching nature.
• Y-coordinate \( \bar{y} \): For the y-coordinate, the formula \( \bar{y}= \frac{1}{2A} \int_{0}^{1} (f(x))^{2} \, dx \) is used. With calculations, it results in \( \bar{y} = \frac{n+1}{2n+2} \). This coordinate is generally located below the midline of the shape's height, illustrating the region's mass distribution.

Using these formulas ensures finding the exact balance points, helping in comprehending the region's shape and size.

Limit Behavior

Exploring the behavior as \( n \rightarrow \infty \) provides insights into how the region evolves. When \( n \) increases indefinitely, the polynomial \( x^n \) transforms significantly:

• The graph flatten towards the \( x \)-axis between \( 0 < x < 1 \), turning into a more exaggerated exponential drop, only touching 1 at \( x = 1 \). Essentially, it becomes a steep wall at \( x = 1 \) as \( n \) grows.
• Geometrically, this shift results in the region appearing to shrink on the \( y \)-axis yet subtly stretches along the \( x \)-axis.

For the centroid, changes occur too:

• The x-coordinate \( \bar{x} \) naturally drifts towards 1, indicating that the center of mass is pulling closer to the far-right boundary as the region becomes more right-leaning.
• The y-coordinate \( \bar{y} \) approaches 0.5, suggesting the vertical balance less reliant on height, aligning with the region's form of a slender line extending upward from the x-axis.

Such limit behaviors offer deep understanding into the dynamics of functions as their parameters expand, a pivotal part of calculus and analysis.

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Key Concepts

Definite Integrals

A definite integral helps us find the area under a curve between two points. It is a fundamental concept in calculus essential for various applications. In this problem, we need to calculate the area under the function \( f(x) = x^n \) from \( x = 0 \) to \( x = 1 \). Here's how it works:

• The integral \( \int_{0}^{1} x^n \, dx \) is computed, resulting in \( \frac{1}{n+1} \). This value represents the total area under the curve, which is crucial for further calculations.
• Definite integrals give you exact numerical values rather than a function, thus allowing you to determine quantities like area with precise bounds.

The role of the definite integral in this exercise provides the baseline for determining the region's centroid.

Centroid Coordinates

The centroid of a region acts similarly to a center of mass, balancing the shape perfectly. To find it, we need coordinates \( \bar{x} \) and \( \bar{y} \). Here’s how each is determined:

• X-coordinate \( \bar{x} \): We use the formula \( \bar{x} = \frac{1}{A} \int_{0}^{1} x \cdot f(x) \, dx \). With the previously found area, \( A = \frac{1}{n+1} \), it evaluates to \( \bar{x} = \frac{n+1}{n+2} \). This coordinate leans towards the higher \( x \) values due to the curve's stretching nature.
• Y-coordinate \( \bar{y} \): For the y-coordinate, the formula \( \bar{y}= \frac{1}{2A} \int_{0}^{1} (f(x))^{2} \, dx \) is used. With calculations, it results in \( \bar{y} = \frac{n+1}{2n+2} \). This coordinate is generally located below the midline of the shape's height, illustrating the region's mass distribution.

Using these formulas ensures finding the exact balance points, helping in comprehending the region's shape and size.

Limit Behavior

Exploring the behavior as \( n \rightarrow \infty \) provides insights into how the region evolves. When \( n \) increases indefinitely, the polynomial \( x^n \) transforms significantly:

• The graph flatten towards the \( x \)-axis between \( 0 < x < 1 \), turning into a more exaggerated exponential drop, only touching 1 at \( x = 1 \). Essentially, it becomes a steep wall at \( x = 1 \) as \( n \) grows.
• Geometrically, this shift results in the region appearing to shrink on the \( y \)-axis yet subtly stretches along the \( x \)-axis.

For the centroid, changes occur too:

• The x-coordinate \( \bar{x} \) naturally drifts towards 1, indicating that the center of mass is pulling closer to the far-right boundary as the region becomes more right-leaning.
• The y-coordinate \( \bar{y} \) approaches 0.5, suggesting the vertical balance less reliant on height, aligning with the region's form of a slender line extending upward from the x-axis.

Such limit behaviors offer deep understanding into the dynamics of functions as their parameters expand, a pivotal part of calculus and analysis.