definite integral
The definite integral is a mathematical concept that represents a specific real number, which is determined by evaluating the integral of a function over a given interval. This real number is interpreted as the net signed area bounded by the function and the x-axis within the specified interval. The integral sign, limits of integration, and integrand are key components that define the structure and meaning of the definite integral.
The definite integral of a continuous function over a specified interval is a real number that quantifies the net signed area bounded by the function and the x-axis within that interval. This value is obtained through a process known as evaluating the definite integral, which involves the integrand, the integral sign, and the limits of integration.
Interpretation
The definite integral represents the net area between a function and the horizontal axis over a specified interval. When the interval has identical start and end points, the integral evaluates to zero since no area is enclosed.
Notation
The definite integral is denoted by the integral sign, with the integrand, limits of integration, and the interval specified in its notation. The referenced quantity represents the real number obtained through the process of evaluating the definite integral.
Conditions and Properties
The definite integral possesses properties analogous to those of finite sums, particularly concerning the treatment of sums and constant multiples of functions. These properties allow the integral of a sum of functions to be expressed as the sum of their integrals and the integral of a constant multiple of a function to be the constant multiple of the integral of the function.
Applications
Definite integrals are used to express the change in position and the distance traveled by a moving object when the velocity function is known. The change in position of the object is given by the definite integral of the velocity function over the interval. Additionally, for any continuous function, the definite integral can be defined as the limit of a Riemann sum, which provides a precise method for calculating the referenced quantity.