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Chapter 7: Problem 57

Finding a Centroid 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?

Short Answer

Expert verified
The centroid of the region bounded by \(f(x)=x^{n},\) the \(x\) -axis, and \(x=1\) is (.\frac{1}{n+2}, \frac{1}{2(n+1)(2n+1)}). As \(n \rightarrow \infty\), the x-coordinate of the centroid approaches 1 and y-coordinate approaches 0.

Step by step solution

01

Find the Area

First, calculate the area under the curve of \(f(x) = x^n\), from 0 to 1. This is given by integration: \(A = \int_0^1 x^n dx = [\frac{x^{n+1}}{n+1}]_0^1 = \frac{1}{n+1}\).
02

Calculate the Centroid

Next, calculate the x and y coordinates of the centroid. The x-coordinate (mean x value) is given by dividing the integral of x times the function by the total area, \(x虅 = \frac{1}{A}\int_{0}^{1}x * x^n dx = \frac{1}{A} [\frac{x^{n+2}}{n+2}]_0^1 = \frac{1}{n+2}\). The y-coordinate (mean y value) is given by dividing the integral of (1/2) times the function squared by the total area, \(y虅 = \frac{1}{2A}\int_{0}^{1} (x^n)^2 dx = \frac{1}{2A} [\frac{x^{2n+1}}{2n+1}]_0^1 = \frac{1}{2(n+1)(2n+1)}\).
03

Evaluate as n approach infinity

As n approaches infinity, this specific curve \(f(x) = x^n\) from x=0 to x=1 resembles a right angle. The centroid moves closer to the x-axis, since the y-coordinate approaches 0. The x-coordinate approaches 1, as n approaches infinity, the vertical line on x=1 dominates the shape of the region.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration
Integration is a fundamental concept in calculus. It refers to the process of calculating the area under a curve between two points. In our exercise, we need to find the area of the region under the curve given by the function \(f(x) = x^n\) from \(x=0\) to \(x=1\). This is accomplished using a definite integral.
  • The definite integral of \(f(x)\) from 0 to 1 is calculated as \(\int_0^1 x^n \, dx\).
  • Using the power rule for integration, we have \(\frac{x^{n+1}}{n+1}\) evaluated from 0 to 1.
  • This results in an area \(A = \frac{1}{n+1}\).
Understanding how to integrate polynomial functions is crucial for finding areas and centroids of regions.
Area under a Curve
The area under a curve provides valuable information about a region bounded by a function and the axes. In this context, calculating the area under the curve \(f(x) = x^n\) from \(x=0\) to \(x=1\) is the first step to find the centroid of the region.
  • For \(f(x) = x^n\), the area gives a measure of the space the curve encloses with the x-axis and the vertical line \(x=1\).
  • The calculated area, \(\frac{1}{n+1}\), changes as \(n\) varies, influencing both the size and the shape of the region.
  • The smaller the area, the steeper the curve toward the vertical line at \(x=1\).
Finding the area is essential as it not only helps in locating the centroid but also provides insight into the geometrical properties of the region.
Centroid Calculation
The centroid, often referred to as the center of mass, is the average location of all the points of an area. Here, to find the centroid of the region defined by \(f(x) = x^n\), we determine the \(x\) and \(y\) coordinates separately.
  • The \(x\)-coordinate of the centroid is found by \(\bar{x} = \frac{1}{A} \int_0^1 x \cdot x^n \, dx\), which simplifies to \(\frac{1}{n+2}\).
  • The \(y\)-coordinate is calculated by \(\bar{y} = \frac{1}{2A} \int_0^1 (x^n)^2 \, dx\), resulting in \(\frac{1}{2(n+1)(2n+1)}\).
  • As \(n\) increases, the centroid's \(x\) and \(y\)-coordinates shift due to the curve's changing geometry.
Locating the centroid gives us a comprehensive understanding of the balance point of the region.
Limit of a Function
Understanding how a function behaves as it approaches a certain value is fundamental in calculus. In this exercise, examining the limit as \(n\) goes to infinity for the function \(f(x) = x^n\) reveals interesting insights.
  • As \(n\) increases indefinitely, \(f(x) = x^n\) approximates a step-like function forming a vertical line at \(x=1\).
  • The region looks increasingly like a rectangle holding a right angle between \(x=1\) and the x-axis.
  • This drastically minimizes the \(y\)-coordinate of the centroid towards zero while the \(x\)-coordinate approaches 1.
Recognizing this limit behavior assists in predicting how the shape and position of the region evolve with large \(n\).

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Most popular questions from this chapter

Astroid Find the area of the surface formed by revolving the portion in the first quadrant of the graph of \(x^{2 / 3}+y^{2 / 3}=4\) , \(0 \leq y \leq 8,\) about the \(y\) -axis.

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Graphical and Numerical Reasoning Consider the region bounded by the graphs of \(y=x^{2 n}\) and \(y=b,\) where \(b>0\) and \(n\) is a positive integer. (a) Sketch a graph of the region. (b) Set up the integral for finding \(M_{y}\) . Because of the form of the integrand, the value of the integral can be obtained without integrating. What is the form of the integrand? What is the value of the integral and what is the value of \(\overline{x} ?\) (c) Use the graph in part (a) to determine whether \(\overline{y}>\frac{b}{2}\) or \(\overline{y}<\frac{b}{2} .\) Explain. (d) Use integration to find \(\overline{y}\) as a function of \(n .\) (e) Use the result of part (d) to complete the table. (f) Find \(\lim _{n \rightarrow \infty} \overline{y}\) (g) Give a geometric explanation of the result in part (f).

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