Definite Integrals Exercises

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.