1. Definite Integral
1. Definition of the Definite Integral
1. If the function is continuous over the interval including two real numbers and , and is an indefinite integral of , then the change in as varies from to , denoted by , is called the definite integral of from to . It is represented symbolically as:
   
   Hence,
   
   This can also be written as:
   
   Threfore,
   To compute the value of the definite integral , we say we are integrating the function from to , where is called the lower limit and is the upper limit. The interval is referred to as the integration interval.
2. If is continuous over an interval including two real numbers and , the following properties hold:


2. Relationship between Integration and Differentiation
   If is continuous on the closed interval , then the definite integral of from to (where ) is:
   
   Therefore, the following holds:
   
   In general, the relationship between integration and differentiation is given by:
   


2. Computation of Definite Integrals
1. Properties of Definite Integrals
   If and are continuous functions on the interval including the three real numbers , , and , the following properties hold:
1. ( is a constant)


2. Functions Defined by Definite Integrals
1. Differentiation of Functions Defined by Definite Integrals
1. ( is a constant)
2. ( is a constant)
2. Limits of Functions Defined by Definite Integrals