The definite integral of a function is closely related to the antiderivative and indefinite integral of a function. The primary difference is that the indefinite integral, if it exists, is a real number value, while the latter two represent an infinite number of functions that differ only by a constant. 

The relationship between these concepts is will be discussed in the section on the Fundamental Theorem of Calculus, and you will see that the definite integral will have applications to many problems in calculus.

Definition of definite integrals
The development of the definition of the definite integral begins with a function f( x), which is continuous on a closed interval [ a, b]. The given interval is partitioned into “ n” subintervals that, although not necessary, can be taken to be of equal lengths (Δ x). An arbitrary domain value, x i, is chosen in each subinterval, and its subsequent function value, f( x i), is determined. The product of each function value times the corresponding subinterval length is determined, and these “ n” products are added to determine their sum. This sum is referred to as a Riemann sum and may be positive, negative, or zero, depending upon the behavior of the function on the closed interval. For example, if f( x) > 0 on [ a, b], then the Riemann sum will be a positive real number. If f( x) < 0 on [ a, b], then the Riemann sum will be a negative real number. 

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