Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

Decide whether the given statement is true or false. Then justify your answer. If \(f\) is continuous and \(f(x) \geq 0\) for all \(x\) in \([a, b]\), then \(\int^{b} f(x) d x \geq 0\)

Step-by-Step Solution

Answer

The statement is true: \( \int_a^b f(x) \, dx \geq 0 \) when \( f(x) \geq 0 \) for all \( x \) in \([a, b]\).

Step 1: Understand the Statement

The problem statement claims that if a function \( f \) is continuous and non-negative on the interval \([a, b]\), then its integral over that interval is also non-negative.

Step 2: Recall Properties of Definite Integrals

The definite integral \( \int_a^b f(x) \, dx \) represents the net area under the curve of \( f(x) \) from \( x = a \) to \( x = b \) on the x-axis. If \( f(x) \geq 0 \) for all \( x \), then the area is above the x-axis and therefore non-negative.

Step 3: Use the Non-negativity of \( f(x) \)

Since \( f(x) \geq 0 \) for all \( x \) in \([a, b]\), whenever \( x \) is within this interval, \( f(x) \) does not fall below zero. Therefore, the expression \( \int_a^b f(x) \, dx \) cannot sum to a negative value because each small slice \( f(x) \cdot dx \) contributes a non-negative quantity.

Step 4: Conclude the Integral's Value

Because every part of the integral contributes non-negatively and the integration is over a closed interval \([a, b]\), where \( f(x) \) is continuous, the integral itself must be non-negative as well. Thus, \( \int_a^b f(x) \, dx \geq 0 \).

Step 5: Final Decision

Based on the analysis and properties of integrals, the statement is true.

Key Concepts

Continuous Functions

In calculus, a function is considered continuous if you can draw it without picking up your pencil. This means there are no gaps, jumps, or holes in the graph of the function at any point within its domain. A continuous function has some important properties, particularly when it comes to integrating them over an interval. For example, if a function is continuous on a closed interval \([a, b]\), it takes on every value between \(f(a)\) and \(f(b)\). This makes it easier to predict the behavior of the function over the interval.

• Continuity ensures integration won't encounter undefined points.
• This smoothness allows for the application of various calculus theorems, such as the Intermediate Value Theorem.
• Most real-world phenomena modeled by continuous functions are predictable and well-behaved within specific ranges.

Understanding the concept of continuity helps in drawing conclusions about the behavior of functions under integration, particularly how they contribute to the overall area calculated.

Definite Integrals

The definite integral of a function on a closed interval, say from \(a\) to \(b\), represents the net area under the curve of a function \(f(x)\) and above the x-axis. It is denoted by \( \int_a^b f(x) \, dx \). Calculating a definite integral involves summing up infinitesimally small pieces of the function's height (\(f(x)\)) times its infinitesimally small width (\(dx\)).

• Definite integrals provide the total sum of these areas between the curve and the x-axis.
• If \(f(x)\) lies above the x-axis, the net area is positive.
• This process essentially gives us a way to accumulate bits of information regarding the changes in function over an interval.

Understanding definite integrals as an area helps us make sense of their value – whether positive or negative – depending on the position of the function relative to the x-axis. A non-negative function ensures that the resulting integral will be non-negative, as each bit of area (even if small) contributes positively.

Non-negative Functions

Non-negative functions are those functions for which the value never falls below zero over a given interval. Mathematically, if \(f(x)\geq0\) for all \(x\) in a specific interval, we treat \(f(x)\) as a non-negative function within that range. It is simpler to deal with such functions because their graph lies entirely on or above the x-axis, eliminating the need to consider any negative contributions to the integral.

• For non-negative functions, the resulting integral or area calculation will always be zero or positive.
• This property simplifies many calculations and predictions about the function's behavior.
• They are particularly important in fields such as probability and statistics, where non-negative distributions ensure coherent, interpretable results.

Consequently, when integrating non-negative continuous functions over specified intervals, the integral's non-negative result follows logically from the properties of both the integration process and the inherent nature of non-negative functions.