Riemann Sums

Definition (Riemann Sum) 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 .

Example (Riemann Sum) Given the function the closed bounded interval and the partition compute a Riemann sum.

    Solution. Organizing into a table we compute the values,  



So the Riemann sum for these is




  



  

Example (Riemann Sum and Area) Use a Riemann sum to approximate the area under the graph of on with 8 subintervals.

    Solution. As a partition we choose, Organizing our computations and choices for our subinterval representations we have,   
    


So the Riemann sum for this partition and these is




  

Example (Riemann Sum and Area) Use a Riemann sum to approximate the area under the graph of on with 10 subintervals.

    Solution. As a partition we choose,
    

    
Organizing our computations and choices for our subinterval representations we have,   
    




  

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