So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and underestimates for decreasing curves.

My question is regarding midpoint approximations of area under a curve for both increasing and decreasing functions. There doesnt seem to be an obvious answer to this without evaluating the integral itself and comparing. So does the midpoint approximation rule over or under estimates a increasing and decreasing function?


The midpoint approximation underestimates for a concave up (aka convex) curve, and overestimates for one that is concave down. There's no dependence on whether the function is increasing or decreasing in this regard.

  • $\begingroup$ So I would have to find the second derivative of the function to see where the over and under estimations? $\endgroup$ – Anson Jan 12 '14 at 23:34
  • $\begingroup$ Yes, the second derivative always positive on the interval (concave up) tells us the midpoint rule will be an underestimate, and the second derivative always negative means the midpoint rule will be an overestimate. $\endgroup$ – hardmath Jan 12 '14 at 23:49
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    $\begingroup$ The trapezoid rule is opposite in this regard, that it overestimates the concave up curves, and underestimates the concave down curves. It's a little easier to visualize that. Combining the midpoint and trapezoid rules in the best way gives Simpson's Rule, which improves the order of accuracy. $\endgroup$ – hardmath Jan 12 '14 at 23:51
  • $\begingroup$ @Zachary F: You attempted to comment on my Answer by editing it. Since you are new here, you lack the minimum reputation needed to Comment on the posts of others. The midpoint rule (and other rules mentioned) are approximations to definite integrals. This is the context in which it makes sense to say that the midpoint rule gives an underestimate for concave up curves (convex functions) and overestimates for concave down curves (concave functions). $\endgroup$ – hardmath Apr 17 at 16:33

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