Question

Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for the integral from 0 to 4 of x squared, dx.

Answer 1

recall the formula for the area of a trapezoid:
A = (B+b)h/2

Its area is the average of the areas of the rectangles formed using the lengths of its bases: the left and right function values!