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