Question

If \(f:[a,b]\to R\) is Riemann integrable, then

• A. \(\left|\displaystyle\int_a^b f(x)\,dx\right|\leq \displaystyle\int_a^b |f(x)|\,dx\)
• B. \(\displaystyle\int_a^b |f(x)|\,dx\leq \left|\displaystyle\int_a^b f(x)\,dx\right|\)
• C. \(\displaystyle\int_a^b |f(x)|\,dx=\left|\displaystyle\int_a^b f(x)\,dx\right|\)
• D. \(\left|\displaystyle\int_a^b f(x)\,dx\right|=\displaystyle\int_a^b f(x)\,dx\)

Hint:

The absolute value of an integral is always less than or equal to the integral of the absolute value.

Answer 1

The Correct Option is A

Concept:
A standard inequality for integrals is \[ \left|\int_a^b f(x)\,dx\right|\leq \int_a^b |f(x)|\,dx \] This is the integral form of the triangle inequality.

Step 1: Use pointwise inequality. For every \(x\), \[ -|f(x)|\leq f(x)\leq |f(x)| \]

Step 2: Integrate throughout. \[ -\int_a^b |f(x)|\,dx \leq \int_a^b f(x)\,dx \leq \int_a^b |f(x)|\,dx \] This implies \[ \left|\int_a^b f(x)\,dx\right| \leq \int_a^b |f(x)|\,dx \]

Step 3: Final answer. \[ \boxed{\left|\int_a^b f(x)\,dx\right|\leq \int_a^b |f(x)|\,dx} \]