Understanding Integration and the Area Under a Curve
Integration is one of the two core operations in calculus, acting as the inverse process of differentiation. While differentiation splits a curve into tiny slopes, integration aggregates those tiny pieces to find the total accumulation, most commonly represented as the area under a curve. Integrals are classified into two categories: indefinite integrals (the general antiderivative formula, written with a "+C" for the unknown constant) and definite integrals (the numeric area bounded between a lower limit a and an upper limit b).
Historically, the concept of integration originated from the method of exhaustion used by ancient mathematicians to calculate the areas of circles and polygons. In the 17th century, the Fundamental Theorem of Calculus formally linked derivatives and integrals, showing that the definite integral of a function can be solved by evaluating its antiderivative at the boundaries. Today, integration is vital for computing volumes, centers of mass, work, probability distributions, and electrical charge accumulations.
This calculator evaluates definite integrals analytically and approximates them numerically using Riemann sums, for three preset functions (a quadratic, sine, and an exponential). By breaking down the integration interval into n subintervals, the solver demonstrates how left, right, and midpoint rectangular approximations converge toward the exact definite integral as n approaches infinity.