Definite integration

 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.
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. The Riemann sum of the function f( x) on [ a, b] is expressed as


  
Certain properties are useful in solving problems requiring the application of the definite integral. Some of the more common properties are
1. 
2. 
3. , where c is a constant
4. 
5. Sum Rule:
6. Difference Rule:
7. If
8. If
9. If
10. If a, b, and c are any three points on a closed interval, then


 


11. The Mean Value Theorem for Definite Integrals: If f( x) is continuous on the closed interval [ a, b], then at least one number c exists in the open interval ( a, b) such that


 


The value of f( c) is called the average or mean value of the function f( x) on the interval [ a, b] and


 

  

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