Question

True or False. Justify your answer with a proof or a counterexample. Assume all functions \(f\) and \(g\) are continuous over their domains. If \(f(x) \leq g(x)\) for all \(x \in[a, b]\), then \(\int_{a}^{b} f(x) \leq \int_{a}^{b} g(x)\).

Short answer

True. If \( f(x) \leq g(x) \) over \([a, b]\), then \( \int_{a}^{b} f(x) \leq \int_{a}^{b} g(x) \) by the properties of definite integrals.

Step-by-step solution

Understanding the Statement

The problem asks us to verify whether the inequality \( \int_{a}^{b} f(x) \leq \int_{a}^{b} g(x) \) holds true if \( f(x) \leq g(x) \) for all \( x \) in the interval \([a, b]\).

Applying the Definition of Definite Integral

The definite integral \( \int_{a}^{b} f(x) \, dx \) represents the net area between the function \( f(x) \) and the x-axis over the interval \([a,b]\). Similarly, \( \int_{a}^{b} g(x) \, dx \) is the net area for \( g(x) \).

Using Properties of Integrals

If \( f(x) \leq g(x) \) for all \( x \in [a, b] \), then the area under the curve \( f(x) \) from \( a \) to \( b \) is less than or equal to the area under the curve \( g(x) \) over the same interval. Thus, \( \int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx \).

Conclusion

Since the inequality \( \int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx \) directly follows from the continuous nature of the functions and the given condition \( f(x) \leq g(x) \), the statement is true.

Key concepts

Continuous Functions

In calculus, a function is said to be continuous if there are no abrupt changes in its value. This means that you can draw the function's graph without lifting your pencil from the paper. Continuous functions are important because they allow us to perform certain operations, like integration over an interval.

When we integrate continuous functions, the result is generally a smooth change, giving us the total area under the curve. This is because continuous functions do not have any gaps or jumps, ensuring a consistent accumulation of values over the interval.

• There are no breaks or interruptions in the graph of a continuous function.
• Every value within the interval contributes to the integral without irregularities.

For our problem, knowing that both functions \(f(x)\) and \(g(x)\) are continuous ensures that integration will not encounter any abrupt spikes or dips, making the use of continuous functions accurate and convenient.

Inequalities

Inequalities tell us about the relative size or order of two values. In our context, we have the inequality \(f(x) \leq g(x)\) for all \(x\) within the interval \([a, b]\). This indicates that the value of function \(f(x)\) does not exceed that of \(g(x)\) for any point in the specified range.

Understanding this inequality is crucial. It allows us to infer that, at every specific point \(x\), "f" lies at or below "g." This direct relationship between the functions helps translate the pointwise inequality into an inequality involving their integrals.

• Inequalities are based on comparing two quantities.
• Ensuring \(f(x) \leq g(x)\) across an interval leads to \(\int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx\).

Such pointwise inequalities are incredibly useful in mathematical proofs and real-world applications, as they form a basis for comparison across entire intervals.

Properties of Integrals

Integrals have several useful properties that make them powerful tools in calculus. A fundamental property is that if one function is always less than or equal to another over a specified interval, the integral of the first function is also less than or equal to the integral of the second function over that interval.

This principle is what the exercise hinges on. If \(f(x) \leq g(x)\) across \([a, b]\), then the total area under \(f(x)\) won't exceed that under \(g(x)\).

• The integral of a smaller function will always be smaller or equal over an interval.
• This property allows us to compare areas directly using integrals.
• The continuous nature of the functions ensures that the property holds true without exceptions.

Collectively, these properties help in verifying and concluding with certainty that \(\int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx\) under the given conditions. This is why it's so vital to understand these integral properties, as they support many mathematical conclusions we encounter.