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Absolute value preserves integrability

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

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

∣∫abf∣≤∫ab∣f∣.\left|\int_a^b f\right|\le \int_a^b |f|.
Remarks

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