Definite Integral Calculator
Numeric evaluation of ∫ₐᵇ f(x) dx by adaptive Simpson quadrature, with the signed area shaded on the graph. Covers §5.15–5.26 of the calculus material.
Definite integral
The definite integral of f from a to b is the signed area between the curve and the x-axis: regions above the axis count positive, regions below count negative. This tool evaluates it numerically with an adaptive method that subdivides the interval more finely wherever the function changes rapidly, and reports the estimate together with its accuracy target.
Evaluate ∫ₐᵇ f(x) dx
Same syntax as the Derivative Engine: sin, cos, tan, ln, exp, sqrt, |x|, pi, e, implicit multiplication.
The curve f(x) with the integration region shaded in gold. FIG. T27-1
Adaptive Simpson works by comparing a coarse Simpson estimate against two half-interval estimates; where they disagree beyond tolerance, the interval is split again. Accuracy concentrates automatically where the curve bends most.
If an endpoint is singular (like 1/√x at 0), the tool nudges the limit inward by a tiny ε and still converges for integrable singularities — but a divergent integral will show unstable, very large values. Watch for that.