Riemann and Darboux Sums

Interactive application for the course MPU1054 – Analysis in One Real Variable II (Chapter 1: the Riemann integral).

This application illustrates the construction of the Riemann integral of a function \( f \) over an interval \( [a, b] \):

• the partition \( \mathcal{P} = \{x_0, x_1, \ldots, x_n\} \) of \( [a, b] \) – uniform, quadratic, random or free – whose points \( x_k \) can be dragged with the mouse and refined with the mouse wheel;
• the tags \( t_k \in [x_{k-1}, x_k] \) (left endpoints, right endpoints, midpoints, random or free) and the Riemann sum \( S(f;\dot{\mathcal{P}}) = \sum_{k=1}^{n} f(t_k)\,(x_k - x_{k-1}) \);
• the lower and upper Darboux sums \( L(f;\mathcal{P}) = \sum_{k=1}^{n} m_k\,(x_k - x_{k-1}) \) and \( U(f;\mathcal{P}) = \sum_{k=1}^{n} M_k\,(x_k - x_{k-1}) \), where \( m_k \) and \( M_k \) are the infimum and the supremum of \( f \) on \( [x_{k-1}, x_k] \), together with the inequalities \( L(f;\mathcal{P}) \le S(f;\dot{\mathcal{P}}) \le U(f;\mathcal{P}) \);
• the limiting process: as the partition is refined, the difference \( U(f;\mathcal{P}) - L(f;\mathcal{P}) \) tends to \( 0 \) and all the sums converge to \( \int_a^b f(x)\,dx \).

Open the application in full screen

Available functions: a non-monotonic “generic” function, \( x \), \( x^2 \), \( x^3 \), \( x^3 - x \), or any expression typed in (for instance sqrt(x) or sin(x) + x/2). The numbers \( a \) and \( b \) and the points \( x_k \) are dragged with the mouse, the mouse wheel changes the number of subintervals, a double-click adds a point to the partition, and the “Animate” button refines the partition automatically. Press ? in the application for the complete help.

Standalone HTML, CSS and JavaScript application in the colours of UQTR, created with the help of Claude Code (Anthropic) from the course notes.