Definite integral

Definition

The definite integral of a function between two points provides the net area under the curve from one point to the other. It is represented by the integral symbol with upper and lower limits.

5 Must Know Facts For Your Next Test

1. The definite integral from $a$ to $b$ of a function $f(x)$ is denoted as $$\int_{a}^{b} f(x) \, dx$$
2. The Fundamental Theorem of Calculus links the definite integral to the antiderivative of a function.
3. The definite integral can be interpreted as the total accumulation of quantities, such as area, volume, or other physical quantities.
4. If $f(x)$ is continuous on $[a, b]$, then the definite integral exists and can be calculated exactly.
5. Properties of definite integrals include linearity, additivity over adjacent intervals, and the effect of reversing limits.

Review Questions

Related terms

Indefinite Integral: An integral without specified limits which represents a family of functions differing by a constant.

Fundamental Theorem of Calculus:

A theorem that establishes the connection between differentiation and integration.

Riemann Sum:

A method for approximating the total area underneath a curve on a graph, often used to define an integral.