Let f be a decreasing function on [a,b] and let R be the region between the graph of f and the x-axis on [a,b].
Will LHS(n) be an over- or under-estimate of the area of R?
Will RHS(n) be an over- or under-estimate of the area of R?
Answer
Whatever shape f has, we know f is decreasing. This means on any sub-interval f will be largest at the left endpoint and smallest at the right endpoint of that sub-interval:
Any left-hand sum will be an over-estimate of the area of R. Since f is decreasing, a left-hand sum will use the largest value of f on each sub-interval. The means any left-hand sum will cover all of R and then some.
Any right-hand sum will be an under-estimate of the area of R. Since f is decreasing, a right-hand sum will use the smallest value of f on each sub-interval. This means any right-hand sum will fail to cover R.
