An integral adds up infinitely many small pieces to find a total. The area under a curve, the distance traveled given a speed, the mass of a variable-density rod—all are integrals. The symbol ∫ is an elongated S, for sum.

There are two kinds. A definite integral gives a number, the accumulated total between two limits. An indefinite integral gives a function, the antiderivative, plus a constant. The fundamental theorem of calculus connects integration to differentiation: they are inverse operations.

Integration is harder than differentiation in practice. Some functions have no elementary antiderivative, so numerical methods are used. Techniques include substitution, integration by parts, and partial fractions. Integrals appear throughout physics, engineering, probability, and economics. They are one of calculus's two central concepts.

• Represents accumulation or area under a curve
• Definite integrals give numbers; indefinite give functions
• Fundamental theorem links it to differentiation
• Often computed numerically