Riemann integral

A Riemann integral of a Riemann integrable function $f:[a,b]\to\mathbb R$ is the number

defined by

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

This definition matches the limit of Riemann sums along partitions of small mesh, and it is the starting point for results such as linearity and the Fundamental Theorem of Calculus .

Examples:

• If $f(x)=c$ is constant on $[a,b]$, then $\int_a^b f(x)\,dx=c(b-a)$.
• For $f(x)=x$ on $[0,1]$, one has $\int_0^1 x\,dx=\tfrac12$.