Absolute value preserves integrability

If a function is Riemann integrable then so is its absolute value, with a triangle inequality.

Absolute value preserves integrability

Absolute value preserves integrability: Let $f:[a,b]\to\mathbb R$ be Riemann integrable on the interval $[a,b]$ . Then the absolute value function $|f|$ is Riemann integrable on $[a,b]$ , and the triangle inequality holds:

\left|\int_a^b f\right|\le \int_a^b |f|.

This is often used together with linearity to control integrals by comparing them to integrals of nonnegative functions.