Exercise 20Calculus › Single-variable calculus › Integration › Riemann integralHigh schoolAdded 5 Nov 2026

Integral of a constant step function

#constant#step function#elementary

Statement

Let $f(x)=c$ for $x\in[a,b]$ . Compute $\int_a^b f$ directly from the definition for step functions.

Hint

Take the trivial subdivision $\{a,b\}$ ; the sum is $c\cdot(b-a)$ .

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Direct computation. With subdivision $\{a,b\}$ , the single interval $[a,b]$ has constant value $c$ . So
$\int_a^b c\,dx=c\,(b-a).$

Conclusion. The integral of a constant equals the constant times the length of the interval.