Question

In this problem we show by examples that the (Riemann) integrability of \(f\) and \(f^{2}\) are independent. (a) Let \(f(x)=\left\\{\begin{array}{ll}{x^{-1 / 2},} & {0

Short answer

Question: Show that the integrability of \(f\) and \(f^2\) are independent when \(f\) is defined as the following functions. (a) \(f(x) = x^{-1/2}\) for \(0 < x \leq 1\) (b) \(f(x) = 1\) if \(x\) is rational and \(f(x) = -1\) if \(x\) is irrational for \(0 \leq x \leq 1\) Answer: (a) We have shown in the solution that the improper integral \(\int_{0}^{1} f(x) dx\) exists but \(\int_{0}^{1} f^2(x) dx\) does not exist. (b) We have shown in the solution that the integral \(\int_{0}^{1} f^2(x) dx\) exists but \(f(x)\) does not have an integral because it is discontinuous at every point in the interval. This demonstrates that the integrability of \(f\) and \(f^2\) are indeed independent.

Step-by-step solution

Compute the integral of \(f(x)\)

For \(0<x\leq 1\), we have \(f(x)=x^{-1/2}\). To show that the integral \(\int_{0}^{1} f(x) dx\) exists as an improper integral, we need to compute its limit: $$\lim_{a \to 0^+} \int_{a}^{1} x^{-1/2} dx$$ This is an improper integral, so we first compute the antiderivative of \(x^{-1/2}\) which is \(2x^{1/2}\), and then calculate the limit.

Evaluate the limit

Now, let's evaluate the limit: $$\lim_{a \to 0^+} \left[ 2x^{1/2} \right]_a^1 = \lim_{a \to 0^+} (2(1)^{1/2} - 2a^{1/2}) = 2-0 =2$$ Since the limit exists, the improper integral \(\int_{0}^{1} f(x) dx\) exists.

Compute the integral of \(f^2(x)\)

For \(0<x\leq 1\), we have \(f^2(x)=x^{-1}\). To show that the integral \(\int_{0}^{1} f^2(x) dx\) does not exist, we need to compute its limit: $$\lim_{a \to 0^+} \int_{a}^{1} x^{-1} dx$$ First, compute the antiderivative of \(x^{-1}\), which is \(\ln(x)\), and then calculate the limit.

Evaluate the limit

Let's evaluate the limit: $$\lim_{a \to 0^+} \left[ \ln(x) \right]_a^1 = \lim_{a \to 0^+} (\ln(1) - \ln(a)) = \lim_{a \to 0^+} -\ln(a)$$ Since the limit does not exist (it goes to infinity), the improper integral \(\int_{0}^{1} f^2(x) dx\) does not exist. ##Part (b)##

Compute the integral of \(f^2(x)\)

Since \(f^2(x)\) would be 1 for all x, we have a simple integral from 0 to 1: $$\int_{0}^{1} f^2(x) dx = \int_{0}^{1} 1 dx$$ We can compute the antiderivative of 1, which is \(x\), and evaluate it from 0 to 1. The integral is \(1-0=1\), so the integral of \(f^2(x)\) exists.

Show that the integral of \(f(x)\) does not exist

The function \(f(x)\) is not Riemann integrable because it is not continuous on any interval, as it takes the values 1 and -1 at every point. When the function is discontinuous at every point in the interval, it is not Riemann integrable. Therefore, \(\int_{0}^{1} f(x) dx\) does not exist. This demonstrates that the integrability of \(f\) and \(f^2\) are indeed independent.

Key concepts

Improper Integrals

Improper integrals are a crucial concept when dealing with certain types of functions where the traditional definite integral cannot be applied directly. These integrals are used when the function has infinite discontinuities or the limits of integration extend to infinity.
To evaluate an improper integral, we often take a limit to handle these complexities. For instance, in the integral \( \int_{0}^{1} x^{-1/2} \, dx \), instead of directly solving from 0 to 1, we examine \( \lim_{a \to 0^+} \int_{a}^{1} x^{-1/2} \, dx \). This approach allows us to bypass the point of discontinuity at \( x = 0 \), ensuring a solution.

• Convert the integral into a limit problem.
• Focus on the antiderivatives during evaluation.
• Approach highlights handling infinite boundaries or singularities.

Properly understanding improper integrals is key to solving problems with limits and discontinuities effectively.

Antiderivatives

Antiderivatives, or indefinite integrals, reverse the process of differentiation. They are essential in evaluating integrals, especially when dealing with improper integrals.

• An antiderivative of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \).
• For example, the antiderivative of \( x^{-1/2} \) is \( 2x^{1/2} \).
• This concept helps convert an integral problem into a simpler evaluation using limits.

In problems involving improper integrals, antiderivatives simplify the calculations.We compute the antiderivative first, then evaluate it through limits.This sequential approach helps handle the infinite behaviors present in certain functions, ensuring we can effectively evaluate their integrals.

Discontinuous Functions

Discontinuous functions pose a challenge for standard Riemann integrability as they might not behave nicely across an interval. When a function isn't continuous over any part of the interval, Riemann integration becomes problematic.

• If a function constantly switches between two values, like 1 or -1, across its domain, it is discontinuous at every point.
• This leads to difficulties in integrating because there is no stable area under the curve that can be quantified.
• Examples include functions that alternate values based on whether \( x \) is rational or irrational.

Despite these challenges, it is essential to recognize how such behaviors impact integrability.Traditional methods may fail, so alternative integration approaches or reviews of the function's behavior are vital for analysis.

Limit Evaluation

Limit evaluation is often necessary when dealing with improper integrals, since we can't evaluate normally at points of discontinuity or boundaries of infinity. By incorporating limits, we work around these points.

• Transition infinite bounds or discontinuous behaviors into a limit expression.
• Handle evaluations by substituting limits that approach the problematic points.
• Observe the results to ascertain if a reasonable finite limit exists.

In the example of \( \int_{0}^{1} x^{-1/2} \, dx \), using \( \lim_{a \to 0^+} (2(1)^{1/2} - 2a^{1/2}) \) let us evaluate around \( x = 0 \) gracefully without encountering undefined solutions.Thus, limit evaluation ensures integral solutions in complex scenarios, crucial for determining the behavior and area under curves that are problematic for traditional methods.