Area Under the Curve Calculator

Formula

How this is calculated

Choose a function f(x) from the list and enter the lower limit a and the upper limit b. The calculator approximates the definite integral ∫_a^b f(x) dx, which represents the signed area between the curve and the x-axis over [a, b].

Integration uses composite Simpson's rule with n = 1000 even subintervals. The step size is Δx = (b − a)/n, and the rule weights the endpoints once, odd-indexed interior samples by 4, and even-indexed interior samples by 2, then multiplies the total by Δx/3. Simpson's rule fits parabolas to pairs of subintervals, giving fourth-order accuracy that is far better than the basic rectangle or trapezoid methods for smooth curves.

The result is a signed area: regions where f(x) is below the x-axis subtract from the total, so the answer can be negative or zero. If a > b the sign flips automatically because Δx becomes negative. Functions like 1/x, √x, and ln(x) are undefined or singular at parts of the real line, so a limit that lands on an undefined point (or an interval crossing such a point) returns no result. The average value is the area divided by (b − a).