Problem 60 - Calculus Early Transcendentals: Pearson New International Edition

Question

Decide whether the given statement is true or false. Then justify your answer. If \(f\) and \(g\) are continuous and \(f(x)>g(x)\) for all \(x\) in \([a, b]\), then \(\left|\int_{a}^{b} f(x) d x\right|>\left|\int_{a}^{b} g(x) d x\right| .\)

Step-by-Step Solution

Answer

True. The absolute values maintain the inequality of the integrals.

Step 1: Analyze the Given Statement

The statement suggests that if there are two continuous functions, \(f\) and \(g\), where \(f(x) > g(x)\) for all \(x\) in the interval \([a, b]\), then the absolute value of the integral of \(f\) over \([a, b]\) is greater than the absolute value of the integral of \(g\) over the same interval.

Step 2: Understand the Properties of Integrals

Recognize that because \(f(x) > g(x)\) for all \(x\) in \([a, b]\), the integrals of these functions over \([a, b]\) should maintain this inequality. Namely: \( \int_{a}^{b} f(x) \, dx > \int_{a}^{b} g(x) \, dx \).

Step 3: Evaluate Absolute Value Considerations

Since both \(f(x)\) and \(g(x)\) are greater than zero and hence positive over \([a, b]\), their integrals will also be positive. Therefore, the absolute values of these integrals are simply the integrals themselves: \( \left| \int_{a}^{b} f(x) \, dx \right| = \int_{a}^{b} f(x) \, dx \) and \( \left| \int_{a}^{b} g(x) \, dx \right| = \int_{a}^{b} g(x) \, dx \).

Step 4: Conclusion on the Truthfulness of the Statement

Since both integrals are positive, the inequality \( \left| \int_{a}^{b} f(x) \, dx \right| > \left| \int_{a}^{b} g(x) \, dx \right| \) directly follows from \( \int_{a}^{b} f(x) \, dx > \int_{a}^{b} g(x) \, dx \). Thus, the statement is True.

Key Concepts

Continuous Functions

Continuous functions are a fundamental concept in calculus that describe functions without any abrupt changes or gaps. In simple terms, a function is said to be continuous over an interval \([a, b]\) if you can draw its graph without lifting your pencil. To understand this concept better, consider the following key points:

• A continuous function has no breaks, jumps, or holes within the interval it's defined on.
• For a function \(f(x)\) to be continuous at a point \(c\), the limit as \(x\) approaches \(c\) from both sides must exist and equal \(f(c)\).
• Any polynomial function is continuous on all real numbers, while other functions like rational functions are continuous wherever their denominators are non-zero.

Continuous functions are essential because they guarantee certain properties when we integrate them, such as the Intermediate Value Theorem and guarantees for existing integrals over closed intervals.

Integral Inequality

An integral inequality essentially involves comparing the integrals of two functions over the same interval. In the problem we are discussing, we have two continuous functions \(f(x)\) and \(g(x)\) where \(f(x) > g(x)\) over the interval \([a, b]\). This means that at every point in the interval, \(f(x)\) is greater than \(g(x)\).Integrating a function over an interval involves calculating the area under its curve from \(a\) to \(b\). Since the entire area under \(f(x)\) is above the area under \(g(x)\) throughout, it is intuitive that:

• The integral of \(f(x)\) over \([a, b]\) is greater than that of \(g(x)\).
• This holds true because each infinitesimally small segment in the integral of \(f(x)\) contributes more area compared to \(g(x)\).
• The inequality follows directly from the property of integrals preserving inequalities when both integrands maintain a consistent difference.

Properties of Integrals

Integration is a powerful tool in calculus with specific properties that make it valuable for analyzing functions. Here are some notable properties:

• Linearity: The integral of a sum of functions equals the sum of their integrals. For example, \(\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx\).
• Monotonicity: If one function is greater than another throughout an interval, its integral will also be greater over that interval, as used in the exercise.
• Additivity Over Intervals: Integrating over separate, adjacent intervals can be summed. For example, \(\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx\).

Understanding these properties helps determine how functions behave over an interval when integrated, as illustrated by the problem. Because \(f(x) > g(x)\) over \[a, b\], we apply the monotonicity of integrals, verifying that the absolute integral of \(f(x)\) will be greater than that of \(g(x)\).