Definite Integral

analysis

Definition

Definite Integral

Let be an interval and a function. If every sequence of Riemann sums, whose corresponding sequence of partitions satisfies , converges to the same limit, then this limit is called the definite integral of on and is denoted by:

Geometrically, for a non-negative function, the definite integral represents the signed area under the curve between the limits of integration and .

Boundary Properties

If the limits of integration are equal or reversed, one defines:

Integration Techniques

Substitution

Let be continuous on , and a differentiable function with and . Then:

Integration by Parts

If and are continuously differentiable on , then: