Numerical Integration with the Midpoint Rule

    The midpoint rule uses a Riemann sum where the subinterval representatives are the midpoints of the subintervals. For some functions it may be easy to choose a partition that more closely approximates the definite integral using midpoints.

Definition (Midpoint Rule) Let be a continuous function on and let



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



where

Example (Midpoint Rule) Consider on and let  



be a partition of the interval (so ). Then the Riemann sum formed by using midpoints 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.

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