Examples for Calculating the Riemann Integral
Use the approximation by step functions to calculate integrals.
Discover Bridges
Study Examples for Calculating the Riemann Integral
We will now give an example of an integral that will be computed according to the definition. This will turn out to be really exhausting even for this quite simple example.
Example 1. Consider the function \(f:[0,1]\to\mathbb{R}\) with \(f(x)=x\). Determine \(\int_0^1f(x)\, dx=\int_0^1x\, dx\). First we consider two sequences of step functions \((\phi_n)_{n\in\mathbb{N}}\), \((\psi_n)_{n\in\mathbb{N}}\) with \[\begin{aligned} \phi_n(x)&=\frac{k-1}n\text{ for }x\in\left[\frac{k-1}n,\frac{k}n\right),\qquad k\in\{1,\ldots,n\},\\ \psi_n(x)&=\frac{k}n\text{ for }x\in\left[\frac{k-1}n,\frac{k}n\right),\qquad k\in\{1,\ldots,n\}. \end{aligned}\] Then for all \(n\in\mathbb{N}\) holds \(\phi_n\leq f\leq \psi_n\). Now we calculate \[\begin{aligned} \int_0^1\phi_n(x)\, dx&=\sum_{k=1}^n\frac{k-1}n\cdot\left(\frac{k}n-\frac{k-1}n\right)\\ &= \sum_{k=1}^n\frac{k-1}{n}\cdot\frac{1}n\\&=\frac{1}{n^2}\sum_{k=1}^n(k-1)\\&=\frac{1}{n^2}\cdot\frac{n(n-1)}2=\frac12-\frac{1}{2n} \end{aligned}\] and \[\begin{aligned} \int_0^1\psi_n(x)\, dx&=\sum_{k=1}^n\frac{k}n\cdot\left(\frac{k}n-\frac{k-1}n\right)\\&= \sum_{k=1}^n\frac{k}{n}\cdot\frac{1}n\\&=\frac{1}{n^2}\sum_{k=1}^nk\\&=\frac{1}{n^2}\cdot\frac{n(n+1)}2=\frac12+\frac{1}{2n}. \end{aligned}\] In particular, we have for all \(n\in\mathbb{N}\) that \[\frac12-\frac{1}{2n}=\int_0^1\phi_n(x)\, dx\leq\int_0^1x\, dx\leq \int_0^1\psi_n(x)\, dx=\frac12+\frac{1}{2n}\] and thus \[\int_0^1x\, dx=\frac12.\]
By the definition of the integral, it is not difficult to obtain that for \(f\in\mathcal{R}([a,b])\) and \(c\in(a,b)\) holds \[\int_a^bf(x)\, dx=\int_a^cf(x)\, dx+\int_c^bf(x)\, dx.\] To make this formula also valid for \(c\geq b\) or \(c\leq a\), we define for \(a\geq b\) that \[\int_a^bf(x)\, dx:=-\int_b^af(x)\, dx.\]
Discuss your questions by typing below.
Solve the Exercise
The data for the interactive network on this webpage was generated with pntfx Copyright Fabian Gabel and Julian Großmann. pntfx is licensed under the MIT license. Visualization of the network uses the open-source graph theory library Cytoscape.js licensed under the MIT license.