Step 1: Understanding the Concept:
This question asks about the relationship between the integral of a function \(f\) and the integral of its absolute value \(|f|\). This is a standard property of Riemann integrals, analogous to the triangle inequality.
Step 2: Key Formula or Approach:
The key property is based on the inequality \(-|f(x)| \le f(x) \le |f(x)|\) for all \(x\).
By the monotonicity property of integrals, if \(g(x) \le h(x)\) for all \(x \in [a,b]\), then \(\int_a^b g(x)dx \le \int_a^b h(x)dx\).
We also need to know that if \(f\) is Riemann integrable, then \(|f|\) is also Riemann integrable.
Step 3: Detailed Explanation:
For any \(x \in [a,b]\), we have the following inequality:
\[ -|f(x)| \le f(x) \le |f(x)| \]
Since \(f\) is Riemann integrable, \(|f|\) is also Riemann integrable. We can integrate this chain of inequalities over the interval \([a,b]\):
\[ \int_a^b -|f(x)|dx \le \int_a^b f(x)dx \le \int_a^b |f(x)|dx \]
\[ -\int_a^b |f(x)|dx \le \int_a^b f(x)dx \le \int_a^b |f(x)|dx \]
This compound inequality is the definition of the absolute value inequality. It can be rewritten as:
\[ \left| \int_a^b f(x)dx \right| \le \int_a^b |f(x)|dx \]
Now let's analyze the options:
- (A) \(\int_a^b f(x)dx \le \int_a^b |f(x)|dx\): This is the right-hand side of our derived inequality. It is always true.
- (B) \(\int_a^b |f(x)|dx \ge |\int_a^b f(x)dx|\): This is the same as the full result we derived, just written in reverse. It is also always true and is the most complete and standard form of the inequality.
- (C) \(\int_a^b f(x)dx = \int_a^b |f(x)|dx\): This is only true if \(f(x) \ge 0\) for all \(x \in [a,b]\). It is not true in general.
- (D) \(|\int_a^b f(x)dx| = \int_a^b f(x)dx\): This is only true if \(\int_a^b f(x)dx \ge 0\). It is not true in general.
Both (A) and (B) are correct statements. However, (B) is the stronger, more precise statement known as the integral version of the triangle inequality. The image shows the checkmark on option (A), which is a weaker but still correct statement. In many contexts, (B) would be considered the "best" answer.
Step 4: Final Answer:
The property of Riemann integrals states that \(|\int_a^b f(x)dx| \le \int_a^b |f(x)|dx\). This implies \(\int_a^b f(x)dx \le |\int_a^b f(x)dx| \le \int_a^b |f(x)|dx\), so statement (A) is true. Statement (B) is also true and is a more complete expression of the property. Given the marking, we select (A).