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Tools / Numerical Analysis

Numerical Integration Playground

See how a function is divided and approximated, and how each numerical rule approaches the integral.

Step 1

Set the function, interval and rule

f(x) =
10

Available: + − * / ^ ( ), pi, e, sin, cos, tan, asin, acos, atan, exp, log, log10, sqrt and abs. Enter multiplication with *.

Presets
Guide

Five approximation rules

Left and right rectangles

Use the function value at the left or right endpoint as each rectangle's height. These simple rules retain a directional bias on a slope.

Midpoint rule

Uses the value at the centre of each subinterval. Its error generally decreases faster than the endpoint rules.

Trapezoidal rule

Connects adjacent sample points with straight lines and sums the resulting signed trapezoidal areas.

Simpson's rule

Fits a quadratic through each group of three points. It is highly accurate for smooth functions and requires an even n.

Values are signed areas. Reversing the limits reverses the sign. The high-accuracy reference is also a numerical estimate, not an analytic exact value. Check results independently for singular, discontinuous or rapidly oscillating functions.