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

• How do you represent the definite integral of a function $f(x)$ from $a$ to $b$?
• What does the Fundamental Theorem of Calculus state about definite integrals?
• List at least two properties of definite integrals.

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.