Riemann integral

If $f:[a,b]\to\mathbb{R}$ is Riemann integrable , its Riemann integral over $[a,b]$ is the common value

where $P$ ranges over all partitions of $[a,b]$, and $L(f,P)$, $U(f,P)$ are the lower and upper sums .

Equivalently, $\int_a^b f$ is the number $I$ such that for every $\varepsilon>0$ there exists $\delta>0$ with

for all tagged partitions $(P,T)$ (this equivalence requires a standard theorem).

The Riemann integral is the classical integral used in elementary calculus and remains useful for continuous and piecewise regular functions.

Examples:

• $\int_0^1 x\,dx=\frac12$.
• If $f(x)=c$ is constant, then $\int_a^b f(x)\,dx=c(b-a)$.
• If $f=\mathbf{1}_{[0,1/2]}$ on $[0,1]$, then $\int_0^1 f(x)\,dx=\frac12$.