Area as the Limit of a Sum

    In this topic we outline and illustrate a general approximation scheme which is a special case of the definite integral (limit of a Riemann sum).

Proposition (Area as the Limit of a Sum) Suppose is continuous and throughout the interval Then the area of the region under the curve over this interval is



where

Example (Area as the Limit of a Sum) Find the exact area under the curve on

    Solution. We will use the formula
    


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













(after algebraic simplification)



Therefore, the exact area under the curve bounded by the lines and is Here is a sketch of the region with the area shaded:

Example (Area as the Limit of a Sum) Find the exact area under the curve on   

    Solution. We will use the formula
    


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









(after algebraic simplification)



Therefore, the exact area under the curve bounded by the lines and is Here is a sketch of the region with the area shaded:

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Area As The Limit Of A Sum
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/area-as-the-limit-of-a-sum.html