Problem 57 - Calculus of a Single Variable

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 \(x^n\), the x-axis, and x=1 is located at (\(\frac{2}{n+2}, \frac{2}{2n+2}\)). As \(n \rightarrow \infty\), both coordinates of the centroid approach 0. Therefore, when n approaches infinity, the centroid of the region will be at the origin.

Step 2: Computing the Integral for Area

Compute the integral using the power rule for integration, yielding A = \(\frac{1}{n+1}\).

Step 3: Finding the x-Coordinate of the Centroid

Then let's find the x-coordinate of the centroid using the formula \(\frac{1}{A}\int_a^b{xdA}\). Substituting and solving the integral we get \(\bar{x}= \frac{2}{n+2}\).

Step 4: Finding the y-Coordinate of the Centroid

Now, to find the y-coordinate of the centroid, use the formula \(\bar{y}= \frac{1}{2A}\int_a^b{f^2(x)dA}\). After substituting and solving the integral, we get \(\bar{y}= \frac{2}{2n+2}\).

Step 5: Limit as n Approaches Infinity

As \(n \rightarrow \infty\), the x-coordinate of the centroid approaches 0 and the y-coordinate also approaches 0. Hence, as n approaches infinity, the centroid of the region is located at the origin

Key Concepts

Integral Calculus

Integral calculus is a branch of calculus that deals with finding the total size, or area, under a curve. It is fundamental in determining the properties of geometric shapes and physical phenomena.

In this exercise, we need to calculate the area under the curve defined by the function \( f(x) = x^n \) from \( x = 0 \) to \( x = 1 \). This calculation involves finding the definite integral of the function over the specified interval. The resulting area helps in determining the centroid of the region, which is the 'center of mass' point of the shape formed between the curve and the axis.

Integral calculus allows us to break down this area calculation into manageable parts, which can then be applied to find more complex geometrical properties.

Power Rule for Integration

The power rule for integration is a straightforward method for finding antiderivatives, allowing us to integrate polynomial functions easily.

Given a function of the form \( x^n \), the integral can be computed using the power rule: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \( C \) is the constant of integration. When dealing with definite integrals, this constant drops out. For our exercise dealing with the region under \( f(x) = x^n \), we integrate over the interval \([0, 1]\).

This gives the area \( A = \frac{1}{n+1} \). The simplicity of the power rule makes it an essential tool in integral calculus, especially for calculating areas and volumes.

Centroid Coordinates

Centroid coordinates are calculated using integration to find the average position of all the points in a shape. It's like finding the balance point of the shape. For planar regions like in this exercise, the centroid coordinates \( \bar{x} \) and \( \bar{y} \) are obtained from specific integrals.

For the x-coordinate, we use the formula: \[ \bar{x} = \frac{1}{A} \int_a^b x \, dA \] Where \( dA \) is the differential area. In our case, after applying the integral, it results in \( \bar{x} = \frac{2}{n+2} \).

Similarly, for the y-coordinate, we have: \[ \bar{y} = \frac{1}{2A} \int_a^b f^2(x) \, dA \] After solving, this gives \( \bar{y} = \frac{2}{2n+2} \).

The coordinates \( (\bar{x}, \bar{y}) \) represent the point where the region could be balanced if it were made of a uniform material.

Limit as n Approaches Infinity

The concept of limits is crucial in calculus, providing insights into the behavior of functions as variables approach particular values. In the context of this exercise, we examine what happens to the centroid coordinates as \( n \) increases indefinitely.

As \( n \to \infty \), calculate the limits of both centroid coordinates. The expression for \( \bar{x} = \frac{2}{n+2} \) approaches 0, as the denominator grows much larger than the constant numerator. Similarly, \( \bar{y} = \frac{2}{2n+2} \) also approaches 0.

This implies that as \( n \) becomes infinitely large, the region tends to a thinner and thinner slice near the y-axis. As such, the centroid, or the "center mass," of this region converges to the origin of the coordinate system, \( (0, 0) \).

This shows how calculus and the concept of limits reveal the behavior of geometrical shapes in dynamic scenarios.