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.
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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., $\int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx = \int_{a}^{b} f(x) \, dx$).
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.