Understanding Integrals
What Is an Integral?
An integral computes the area under a curve. For a function f(x), the definite integral from a to b gives the signed area between the curve and the x-axis. The indefinite integral (antiderivative) F(x) satisfies F'(x) = f(x).
The Power Rule for Integration
For a polynomial term axⁿ, the antiderivative is a·xⁿ⁺¹/(n+1) + C. This is the power rule for integration and is the most fundamental integration technique. The constant C represents the family of antiderivatives.
Definite vs Indefinite Integrals
A definite integral ∫ₐᵇ f(x)dx produces a number: the net signed area. An indefinite integral produces a function family. The Fundamental Theorem of Calculus connects them: ∫ₐᵇ f(x)dx = F(b) − F(a).
Riemann Sums
Riemann sums approximate the definite integral by dividing [a,b] into rectangles. The sum of rectangle areas converges to the integral as the number of rectangles grows. This calculator uses right-endpoint Riemann sums for the table visualization.
Applications
Integrals compute areas, volumes of revolution, arc lengths, work done by forces, center of mass, probability distributions, and accumulated change. They are indispensable in physics, engineering, economics, and statistics.