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Definition

The definite integral is a mathematical concept that calculates the accumulated area under a curve within a specified interval on the x-axis. It provides the net area, considering both positive and negative areas.

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 links differentiation and integration, stating that if $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.
3. Definite integrals can be evaluated using numerical methods such as the Trapezoidal Rule and Simpson's Rule when an analytic solution is difficult.
4. When calculating the area between two curves, set up the integral by subtracting the lower function from the upper function within the given interval.
5. A definite integral can represent physical quantities such as area, volume, displacement, and total accumulated change.

Review Questions

Related terms

Indefinite Integral: A type of integral that represents a family of functions and includes an arbitrary constant. It is denoted by $\int f(x) \, dx = F(x) + C$.

Fundamental Theorem of Calculus:

A theorem that establishes the relationship between differentiation and integration. Part 1 relates antiderivatives to definite integrals; Part 2 provides a method to evaluate definite integrals.

Numerical Integration:

Techniques used to approximate the value of definite integrals when analytic solutions are difficult or impossible. Common methods include Trapezoidal Rule and Simpson's Rule.