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.