Numerical Integration Calculator

This calculator finds the value of a definite integral, the area under a curve between two limits, for any function, using Simpson's rule, a fast and accurate numerical method. Integration is a cornerstone of calculus, but many functions cannot be integrated with a tidy formula, so numerical methods estimate the answer by sampling the function and summing the pieces. Simpson's rule is one of the best simple methods: rather than approximating the curve with straight lines, as the trapezoidal rule does, it fits little parabolas through groups of three points, which captures curvature and gives excellent accuracy for a modest number of intervals. It is exact for any polynomial up to cubics and very close for almost everything else. This tool applies it to your own function. You enter the function f(x) in ordinary mathematical syntax, the lower and upper limits of integration, and the number of intervals to use, and the calculator returns the estimated value of the integral, the method, the number of intervals, and the range. More intervals give a more accurate result. The results update as you type. Use it to evaluate integrals that have no closed form, to find the area under a curve or between limits, to compute work, probability or other accumulated quantities, or to check a hand calculation. A few notes: the number of intervals is made even, as Simpson's rule requires, and functions can include powers, roots, exponentials, logarithms and trigonometric functions. For very wiggly functions, increase the number of intervals to improve accuracy. This complements the symbolic integral calculator, which finds an exact antiderivative where one exists, while this tool gives a reliable numerical value for any function at all.

Simpson's rule with an even number of intervals. Exact for polynomials up to cubic; very accurate otherwise. Use JS syntax (x*x, sqrt(x), sin(x)). More intervals = more accuracy.