MHB How can the lower rectangle method for area approximation be improved?

[rsyed5]

Hi,
Can someone suggest a way to improve the lower rectangle method for area approximation...? I know one way is to increase the number of rectangles so if I put 100 instead of 5 rectangles i will get a better approximation. Another way it could be improved...?

 

[MarkFL]

There is the midpoint rule, the trapezoidal rule, Simpson's rule, and higher order Newton-Cotes formulas. If you are in a calculus course, all of these will be introduced to you soon, except perhaps for the Newton-Cotes formulas.

 

[Prove It]

As Mark has suggested, the best way to improve on the Left Hand Estimate is to not use it, but to use a more accurate method. Better yet perform the definite integral to get an exact answer. If worse comes to worse, do a Right Hand Estimate as well and average the results.

 

[HallsofIvy]

In the case that a function is increasing (of decreasing) a "left hand" approximation is the same as your "lower rectangle" approximation and a "right hand" approximation is the same as an "upper rectangle" approximation. The average of the two, as Prove It suggests, is the same as the "trapezoid" method. I believe that "Simpson's rule" is the most accurate of the elementary methods for the same amount of work.