Let \(\phi\in T[a,b]\) be a step function with respect to the partition \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\), i.e. let \(\phi\) take some constant values \(c_i\) on the open intervals \((x_{i-1},x_i)\) \[\phi (x)=c_i\quad\quad\text{for all } x\in(x_{i-1},x_i),\quad i=1,\ldots,n.\] The Riemann integral of the step function is defined as the sum. \[\int_a^b\phi(x)dx:=\sum_{i=1}^nc_i(x_i-x_{i-1}).\]
The Riemann integral of the step function is well-defined, i.e. its definition does not depend on the specific choice of the partition \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\).
The Riemann integral can be interpreted as the area between the \(x\)-axis and the step function. The area is counted positive, if it is above the \(x\)-axis (green) and negative if it is below the \(x\)-axis (red):
Proofs: 1