Question

Function f(x) is positive, increasing and concave down on the closed interval [a, b]. The interval [a, b] is partitioned into 4 equal intervals and these are used to compute the left sum, right sum, and trapezoidal rule approximations for the value of the integral from a to b of f of x dx. Which one of the following statements is true

A. Trapezoidal rule value < Left sum < Right sum
B. Left sum < Trapezoidal rule value < Right sum
C. Right sum < Trapezoidal rule value < Left sum
D. Cannot be determined without the x‐values for the partitions

Answer 1

B. Left sum < Trapezoidal rule value < Right sum

Explanation:

The left sum makes use of the left endpoint of the sub-interval making it less than the right sum which makes use of the right endpoint of the sub-interval.

The trapezoidal approximation makes use of both endpoints, but has an area that is the average of the areas of the inscribed and circumscribed rectangles.

.Hence the trapezoidal rule value will be approximately equal to the average of the right sum and the left sum