A Calculus IV tool
Interactive laboratory of Darboux sums
Build partitions of the unit square, compare lower and upper sums, and observe how they change under refinement. The application supports mouse and keyboard on desktop computers and touch gestures on phones and tablets.
When Q refines P
\[ L(f,P)\leq L(f,Q)\leq U(f,Q)\leq U(f,P). \]
All computations run locally in the browser.
01
Dyadic refinement
Visualize the lower- and upper-sum prisms for partitions with \(n=2^k\) divisions per axis and follow their approximation to the integral.
02
Compare P and Q
Draw two partitions, enter their cuts manually, and compare their four Darboux sums and enclosing intervals at the same time.
03
Sequence P → Q
When \(Q\) refines \(P\), build a chain that adds one cut at each step and animate the monotonicity of the sums.
Laboratory
Accuracy of the sums
The built-in examples use exact formulas for the infimum and supremum on every subrectangle. For a user-entered function, cell extrema are approximated on a finite common grid and marked with \(\approx\).
Proper and improper integrals
The plots can show a known exact value, a user-provided reference, or a numerical cubature. The singularity diagnosis is informative and does not replace a proof.
Installable and offline
After the first visit, compatible browsers can use the Install button to add the laboratory as a web app and keep its files available offline.