Problem 63 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f\) be continuous on \([a, b]\) and thus integrable there. Show that $$ \left|\int_{a}^{b} f(x) d x\right| \leq \int_{a}^{b}|f(x)| d x $$

Step-by-Step Solution

Answer

By the triangle inequality for integrals, \(\left|\int_{a}^{b} f(x) \,dx \right| \leq \int_{a}^{b} |f(x)| \,dx\) holds.

Step 1: Understand the Problem

We are given a function \(f\) that is continuous on the interval \([a, b]\). We need to prove that the absolute value of the integral of \(f\) over \([a, b]\) is less than or equal to the integral of the absolute value of \(f\) over the same interval.

Step 2: Apply the Triangle Inequality for Integrals

Recall that the triangle inequality for integrals states: \(\left| \int_{a}^{b} g(x) \, dx \right| \leq \int_{a}^{b} |g(x)| \, dx\) for any integrable function \(g(x)\) over \([a, b]\). In our case, \(g(x) = f(x)\).

Step 3: Use Properties of Absolute Value

For any real number \(x\), the absolute value \(|x|\) is defined as \(x\) if \(x \geq 0\) and \(-x\) if \(x < 0\). This property allows us to handle the absolute values within the integral.

Step 4: Apply the Property to the Given Function

Let \(g(x) = f(x)\) as mentioned before. Then the property \(\left| \int_{a}^{b} f(x) \, dx \right| \leq \int_{a}^{b} |f(x)| \, dx\) directly follows from the triangle inequality for integrals, since it holds for any integrable function \(f(x)\).

Step 5: Conclude the Proof

We conclude that by the triangle inequality for integrals, the given inequality \(\left| \int_{a}^{b} f(x) \, dx \right| \leq \int_{a}^{b} |f(x)| \, dx\) holds true for any continuous function \(f(x)\) on \([a, b]\), proving the statement.

Key Concepts

Continuous Functions

A continuous function is a type of function where the graph is unbroken, meaning there are no jumps or holes. It smoothly connects every point over its domain. This characteristic is crucial for making sure certain limits and integrals exist and behave predictably. In our exercise, we base the solution on the idea that the function \( f \) is continuous on the interval \([a, b]\) which guarantees that we can integrate it over this interval.

Key insights about continuous functions include:

• They do not have sudden jumps or interruptions in their domain.
• For a function \( f \) to be integrable over an interval \([a, b]\), it must be continuous on that interval.
• Continuity ensures that you can apply the Fundamental Theorem of Calculus effectively.

In our problem, because \( f \) is continuous on \([a, b]\), this ensures that \( f \) is also integrable there, allowing us to work with its integral smoothly.

Absolute Value

The concept of absolute value is a way of measuring the magnitude of real numbers without regard to their direction along the number line. It is particularly useful when dealing with integrals because it helps manage positive and negative values. The absolute value of a number, \(|x|\), is defined as:

• \(x\) if \(x \geq 0\)
• \(-x\) if \(x < 0\)

This property of absolute value enables us to treat integrals and sum up the 'size' or 'magnitude' of functions over an interval.

In the context of the given exercise, the absolute value is important when stating that \( \left| \int_{a}^{b} f(x) \, dx \right| \leq \int_{a}^{b} |f(x)| \, dx \). This inequality uses absolute values to express that the total 'net' effect of \(f(x)\) over the interval \([a, b]\) cannot exceed the total magnitude described by the absolute values.

Triangle Inequality

The Triangle Inequality is a fundamental concept in mathematics that, at a high level, relates the lengths of sides of a triangle. In its application to integrals, it provides us with a way to compare the magnitudes of integral expressions. Specifically, for integrable functions, the Triangle Inequality can be stated as:

• For any integrable function \(g(x)\), the inequality \( \left| \int_{a}^{b} g(x) \, dx \right| \leq \int_{a}^{b} |g(x)| \, dx \) holds.

In simple terms, the absolute value of an integral is always less than or equal to the integral of the absolute values of the function.

This inequality reflects the idea that if you think of the integral as a sum, the total sum's 'net value' can be less than the sum of absolute values since negative contributions can 'cancel out' positive ones. In our proof, the triangle inequality is a major stepping stone to showing that \( \left| \int_{a}^{b} f(x) \, dx \right| \leq \int_{a}^{b} |f(x)| \, dx \). It's the key tool that connects our function analyses to its integral properties.

Integrable Functions

Finally, when we talk about integrable functions in calculus, we refer to functions that can be measured via integration. For a function to be integrable, it usually needs to be bounded and the area under its curve within an interval must be finite. Crucially, continuous functions on a closed interval \([a, b]\) are guaranteed to be integrable over that interval.

Some key points include:

• A function is integrable over \([a, b]\) if you can compute a definite integral from \( a \) to \( b \).
• Integrable functions can be complex or simple, as long as they meet certain conditions, such as being bounded.
• The concept of integrability ensures that the tools of calculus, like the Fundamental Theorem, can be applied.

In our exercise, because \( f \) is continuous on \([a, b]\), it is integrable, which means we can explore its integral and apply inequalities such as the triangle inequality effectively. Thus, integrability is a foundational property that allows us to solve problems involving integrals.