Problem 44 - Essential Calculus

Question

Given continuous functions \(f\) and \(g\) such that \(0 \leq f(x) \leq g(x)\) on the interval \([a, \infty),\) prove the following. (a) If \(\int_{a}^{\infty} g(x) d x\) converges, then \(\int_{a}^{\infty} f(x) d x\) converges. (b) If \(\int_{a}^{\infty} f(x) d x\) diverges, then \(\int_{a}^{\infty} g(x) d x\) diverges.

Step-by-Step Solution

Answer

In this exercise, the comparison test was used to understand the convergence and divergence of two functions \(f\) and \(g\). For a given interval \([a, \infty)\), if the integral \(\int_{a}^{\infty} g(x) d x\) converges, the integral \(\int_{a}^{\infty} f(x) d x\) also converges. Conversely, if the integral \(\int_{a}^{\infty} f(x) d x\) diverges, the integral \(\int_{a}^{\infty} g(x) d x\) also diverges assuming \(0 \leq f(x) \leq g(x)\).

Step 1: Part (a) Step 1: Understand the Condition

This part of the exercise states that if the improper integral of function \(g(x)\) converges, then \(\int_{a}^{\infty} f(x) d x\) also converges. First, it is important to remember that \(0 \leq f(x) \leq g(x)\). By this condition, \(f(x)\) is less than or equal to \(g(x)\) for all \(x\) in the interval \([a, \infty)\). The inequality is important for the argument, as it is needed to apply the comparison test.

Step 2: Part (a) Step 2: Apply Comparison test

Using the conditions met by \(f\) and \(g\), it can be stated that if \(\int_{a}^{\infty} g(x) d x\) converges, so does \(\int_{a}^{\infty} f(x) d x\). This is directly from the comparison test, which states that if \(0 \leq f(x) \leq g(x)\) and the integral from \(a\) to infinity of \(g(x)\) dx is finite (converges), then the integral from \(a\) to infinity of \(f(x)\) dx is also finite (converges).

Step 3: Part (b) Step 1: Understand the Condition

This part of the exercise states that if the improper integral of function \(f(x)\) diverges, then \(\int_{a}^{\infty} g(x) d x\) also diverges. Again, remember the condition \(0 \leq f(x) \leq g(x)\). It means that \(g(x)\) is more than or equal to \(f(x)\) for all \(x\) in the interval \([a, \infty)\). This inequality will form the basis of the comparison test application.

Step 4: Part (b) Step 2: Apply Comparison Test

By the conditions satisfied by \(f\) and \(g\), the statement that if \(\int_{a}^{\infty} f(x) d x\) diverges, then \(\int_{a}^{\infty} g(x) d x\) also diverges, can be proven. Since \(0 \leq f(x) \leq g(x)\), the comparison test states that if the integral from \(a\) to infinity of \(f(x)\) dx is infinite (diverges), then the integral from \(a\) to infinity of \(g(x)\) dx is also infinite (diverges).

Step 5: Summary

By using the comparison test, it was possible to prove both parts of the exercise. The first part proves that if \(g(x)\) converges, then \(f(x)\) will also converge. The second part proves that if \(f(x)\) diverges, \(g(x)\) will also diverge. The core point taken from this exercise is how the comparison test can be used to compare the convergence and divergence of two separate functions when one function is less than or equal to the other function.

Key Concepts

Improper Integrals

Improper integrals are a type of integral which extends to infinity or involves function discontinuity. These integrations pose challenges since they include values that aren't finite, which typically means standard integration techniques don't apply. When you are faced with an improper integral, such as \( \int_{a}^{\infty} f(x) dx \), you're trying to comprehend the area under the curve from a certain point \(a\) to infinity.

Improper integrals determine whether the infinite area under a curve has a finite value, known as convergence, or whether it extends without bounds, known as divergence. These concepts are crucial in understanding the behavior of functions over long intervals and are applied in physics, engineering, and probability theory to model systems over long durations or distances.

Convergence and Divergence

In calculus, convergence and divergence refer to the behavior of an improper integral or an infinite series. When evaluating an improper integral, if the limit of the area under the curve as it approaches infinity is a finite number, the integral is said to converge. Conversely, if the area grows without bound, then the integral diverges.

To assess convergence or divergence without actually finding the exact value, we use comparison tests. These are powerful tools allowing us to compare a challenging-to-evaluate integral to a well-understood integral. If the simpler integral has a known behavior, it informs us about the behavior of the more complex function's integral. For instance, in our given problem when the integral of \( g(x) \) is known to converge, and given that \( 0 \leq f(x) \leq g(x) \), it logically follows that the integral of \( f(x) \) must also converge since it is 'smaller' or equal at any point.

Continuous Functions

Continuous functions, like the ones in our exercise, behave nicely in the realm of calculus. These functions have no gaps, jumps, or points of discontinuity within their domain. This smoothness means you can draw the function's graph without lifting your pencil. The importance of a function being continuous cannot be overstated, particularly when discussing improper integrals.

Continuous functions guarantee that within the interval of interest, the function will not behave erratically, allowing us to rely on integration techniques to evaluate areas under curves. For instance, when the exercise mandates that \(f\) and \(g\) are continuous, we can infer that any potential convergence or divergence is not due to any discontinuities but because of the behavior of the functions at infinity.