Numerical Integration Uisng Left and Right Endpoints

    The left and right rules are the most straight-forward to learn. They can be applied with either uniform width subintervals or varying width subintervals. They consist of using a Riemann sum where the subinterval representatives are chosen as the left-endpoints or the right-endpoints, respectively.  

Definition (Left Rule and the Right Rule) Let be a continuous function on and let



be a partition of the interval Then the Riemann sum formed by using left-endpoints as the subinterval representatives is



and the Riemann sum formed by using right-endpoints as the subinterval representatives is



where in both cases The first formula is called the left rule and the second formula the right rule.

Example (Left Rule and the Right Rule) Consider on and let   be a partition of the interval (so ). Then the Riemann sum formed by using left-endpoints as the subinterval representatives is





The Riemann sum formed by using right-endpoints as the subinterval representatives is





Consider that as increases then so does the estimation of the area. Note that a somewhat accurate estimation is   
    However, because a fairly large number of rectangles are needed for a good approximation there are other common techniques which do not use rectangles.

• print this page
• pages linking here
• related pages

Cite this as:
Numerical Integration Using Left And Right Endpoints
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/numerical-integration-using-left-and-right-endpoints.html