Riemann Sums and Area

    In this topic we illustrate how a Riemann sum can be used to approximate the area under a curve and in doing so, we anticipate the notion of definite integral. We will investigate the area under the curve above the -axis and between the vertical lines and Here is a sketch with the region shaded:



Next let's recall the definition of a Riemann sum.

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 and the are called the subinterval representatives.

    For our first example we will use a partition with low cardinality, say a partition from using only 4 subintervals between and Also, we will use left-endpoints for our subinterval representatives. Our initial estimate for the area is 20 as shown:

Example (Riemann Sums and Area) Given the function the closed bounded interval and the partition compute a Riemann sum to approximate the area.

    Solution. Organizing into a table we compute the values,  



So the Riemann sum for these is


  




The following sketch shows the Riemann sum as the approximate area under the given curve.



  

    For our second example we will use a finer partition, say a partition using 14 subintervals between and (also with uniform width). We will still use left-endpoints for our subinterval representatives. Our second estimate for the area is as shown:

Example (Riemann Sums and Area) Given the function the closed bounded interval and the partition



compute a Riemann sum to approximate the area.

    Solution. Organizing into a table we compute the values,  



So the Riemann sum for these is


  




The following sketch shows the Riemann sum as the approximate area under the given curve.


  

    Here is an animation that demonstrates what's going on:

    
    

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Cite this as:
Riemann Sums And Area
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/riemann-sums-and-area.html