Exercise 48Calculus › Single-variable calculus › Integration › Riemann integralHigh schoolAdded 5 Nov 2026

Monotonicity of the integral

#monotonicity#step functions#comparison

Statement

Let $f,g$ be step functions on $[a,b]$ with $f\le g$ . Show that $\int_a^bf\le\int_a^bg$ .

Hint

Compare the two sums on a common refinement.

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Common refinement. Choose a common subdivision $\sigma$ of $[a,b]$ . On each $]x_{k-1},x_k[$ , $f\le g$ implies $c_k\le d_k$ .

Sum. $\sum c_k\Delta_k\le\sum d_k\Delta_k$ .

Conclusion. The integral is monotone.