Definite Integral

    Recall a Riemann sum for a function on the closed bounded interval is a sum of the form



where and for,

    and     

The set is called a partition of and the largest of the is called the norm of and is denoted by

Definition (Definite Integral) If is defined on the closed interval and if



exists, then this limits is called the definite integral of from to The definite integral is denoted by

    The function that is being integrated is called the integrand; the interval is the interval of integration; and the endpoints and are called, respectively the lower and upper limits of integration.  

Proposition (Definite Integral of a Continuous Function) If is a continuous function on an interval then is integrable on

Example (Evaluating a Definite Integral using the Definition) Evaluate



using the definition of the definite integral.

    Solution. We will use a formula based on equal width subintervals and right endpoints,  
    


with and We see and we notice that as and so the definte integral is given by
    
    











(after algebraic simplification)

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