Problem 59 - Calculus Early Transcendentals

Question

Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\) Let \(A_{n}\) be the area of the region bounded by \(f(x)=x^{1 / n}\) and \(g(x)=x^{n}\) on the interval \([0,1],\) where \(n\) is a positive integer. Evaluate \(\lim _{n \rightarrow \infty} A_{n}\) and interpret the result.

Step-by-Step Solution

Answer

Answer: As \(n\) approaches infinity, the area of the region bounded by the given functions on the interval \([0,1]\) approaches 1. This means that the two functions become more and more similar in shape on the given interval as \(n\) increases, and the area between them tends to 1.

Step 1: Finding the area \(A_n\)

To find the area of the region bounded by the functions, we should subtract \(g(x)\) from \(f(x)\) and then integrate it over the given interval \([0,1]\). $$A_n = \int_{0}^{1}\left( f(x) - g(x) \right) dx = \int_{0}^{1}\left( x^{\frac{1}{n}} - x^{n} \right) dx$$

Step 2: Performing the integration

Now, we need to perform the integration and simplify the result: $$A_n = \left[ \frac{n}{n+1}x^{\frac{n+1}{n}} - \frac{1}{n+1}x^{n+1} \right]_{0}^{1} = \frac{n}{n+1} - \frac{1}{n+1}$$

Step 3: Finding the limit of \(A_n\) as \(n \rightarrow \infty\)

Next, we will evaluate the limit of the area as n approaches infinity: $$\lim_{n \rightarrow \infty} A_n = \lim_{n \rightarrow \infty} \left( \frac{n}{n+1} - \frac{1}{n+1} \right)$$ To evaluate this limit, we can use the fact that as \(n\) becomes very large, the terms \(1/n\) become insignificant compared to \(1\). Thus, the limit becomes: $$\lim_{n \rightarrow \infty} A_n = \lim_{n \rightarrow \infty} \left( 1 - \frac{1}{n+1} \right) = 1$$

Step 4: Interpreting the result

The result of the limit means that, as \(n\) becomes larger and larger, the area of the region bounded by the given functions on the interval \([0,1]\) approaches 1. This implies that the two functions become more and more similar in shape on the given interval as \(n\) increases, and the area between them tends to 1.

Key Concepts

Integration

Integration is a fundamental concept in calculus that is used to find the area under a curve. Here, to find the area between two curves, we subtract one function from the other and integrate that result over the specified interval.
In this exercise, our goal is to determine the area between the curves defined by the functions \(f(x) = x^{1/n}\) and \(g(x) = x^n\) within the interval \([0,1]\).
The operation is expressed as:

• Integrate: \( \int_{0}^{1} (x^{1/n} - x^n) \; dx \).
• This subtraction indicates finding the difference in heights of the functions at every point over the interval.
• The integral accumulates these differences to give the total area between the curves.

Limits

Limits help us understand the behavior of functions as they approach certain values. In this context, limits provide insight into how the area between our two functions changes as the parameter \(n\) becomes very large.
The limit we evaluate is:

• \( \lim_{n \to \infty} A_n = \lim_{n \to \infty} \left( \frac{n}{n+1} - \frac{1}{n+1} \right) \)
• As \(n\) increases, the fractions \(1/n\) and \(1/(n+1)\) approach zero.
• Hence, the expression simplifies to \(1 - 0 = 1\).

This result means that as \(n\) grows, the area between the curves reaches its maximum possible value of 1.

Calculus

Calculus is an area of mathematics that provides tools such as integration and differentiation. It allows us to find rates of change and areas under curves. In this problem, we apply calculus methods to determine how two curves diverge.
The use of integration here stems from calculus, where we aim to understand the accumulation of quantities, like area. Calculus also aids in simplifying the problem by providing the tools to calculate limits, allowing us to predict behavior as \(n\) becomes very large.
By calculating the integral and then finding the limit, we are harnessing calculus to make sense of this mathematical behavior across different values of \(n\).

Functions

Functions are essential mathematical constructs that describe relationships between sets of numbers. The function \(f(x) = x^{1/n}\) implies a root function whose shape changes as \(n\) varies, while \(g(x) = x^n\) represents a power function. The way each of these functions behaves is important to determine how they interact over the given interval.
As \(n\) increases:

• \(f(x)\) becomes more linear, approaching the function \(f(x) = x\).
• \(g(x)\) becomes very rapidly increasing for values close to \(x=1\), and flattening toward \(x=0\).

The behavior of these functions and their graphical representation helps us visualize and understand the concept of the area between the curves. Understanding how these functions change with \(n\) provides insight into why the limit of the area is 1.