Math · Calculus · Concept
From rectangles to an integral
Signed sums approximate accumulation by adding contributions over small intervals.
Height times width
Split [a,b] into n equal pieces. Choose a left endpoint, right endpoint, or midpoint height in each. Multiply by Δx and add. Rectangles below the axis contribute negatively; geometric area would instead use |f|.
Different samples, same limiting target
Continuous functions on a closed interval are Riemann integrable. As the largest partition width tends to zero, these sums approach one signed integral. A finite sample can miss a narrow feature, so a smooth picture or two agreeing sums is not a proof of convergence.
Trapezoids use both endpoint heights
Each trapezoid replaces a curve segment by its secant. This is a useful quadrature rule, though its error depends on the function. The visualizer compares finite sums with a numerical reference estimate, not an exact certificate.
Common mistakes
- Unsigned geometric area differs from signed accumulation.
- Increasing n need not make every approximation error decrease monotonically.
Work through an example
Use two rectangles and compare with the integral.
Compare sums for x² on [0,1] →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.