Question

If the function $f(x)$ is bounded and integrable on $[a,b]$ such that $f(x) \ge 0$ $\forall x \in [a,b]$, $b \ge a$ then $\int_{a}^{b} f(x) dx$ is __}

• A. $\le 0$
• B. $= 0$
• C. $\ge 0$
• D. $x \ne 0$

Hint:

Integration preserves inequalities: if $f(x) \ge g(x)$, then $\int f \ge \int g$.

Answer 1

The Correct Option is C

Step 1: Concept
This refers to the monotonicity property of the Riemann integral.

Step 2: Meaning
If the integrand $f(x)$ is non-negative throughout the interval $[a,b]$, the accumulation of its values (the integral) must also be non-negative.

Step 3: Analysis
Since $f(x) \ge 0$ for all $x$ and the interval length $(b-a)$ is non-negative, every sum used to define the integral is $\ge 0$.

Step 4: Conclusion
Therefore, the limit of these sums, $\int_{a}^{b} f(x) dx$, must be greater than or equal to 0. Final Answer: (C)