Definite integral

Definition

A definite integral is the evaluation of the integral of a function over a specific interval, yielding a real number that represents the net area under the curve between two points.

5 Must Know Facts For Your Next Test

1. The notation for a definite integral is $\int_{a}^{b} f(x) \, dx$, where $a$ and $b$ are the limits of integration.
2. The Fundamental Theorem of Calculus connects definite integrals with antiderivatives, stating that if $F$ is an antiderivative of $f$, then $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$.
3. Definite integrals can be used to calculate areas under curves, volumes of solids of revolution, and other physical properties like work and distance.
4. If the function $f(x)$ is continuous on the interval [a, b], then its definite integral exists.
5. Properties of definite integrals include linearity (e.g., $\int_{a}^{b} [cf(x) + g(x)] \, dx = c \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx$) and additivity over adjacent intervals (e.g., $\int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx = \int_{a}^{b} f(x) \, dx$).

Review Questions

• What does the Fundamental Theorem of Calculus state about definite integrals?
• How would you find the area between two curves using definite integrals?
• What are some properties of definite integrals that simplify their computation?

Related terms

Integral: A general concept in calculus representing accumulation or area under a curve. It can be indefinite or definite.

Antiderivative: A function whose derivative is the given function. It is used in calculating indefinite and definite integrals.

Fundamental Theorem of Calculus:

A theorem that links differentiation with integration, providing a method to evaluate definite integrals using antiderivatives.