1. The graph below is the graph of a function f(x) on the domain [1,3].

The Riemann sums for the left endpoint, right endpoint, midpoint, and trapezoidal sums are, in some order: 6.208852, 6.603778, 6.600176 , 6.998705. (All are for the same N)

(a) Identify each sum which each number.

(b) Does the trapezoidal sum overestimate or underestimate the integral ? Why?

Solution

(a) Since the function is increasing on its domain, the left enpont sum will be minimal, the roght endpoint sum will be maximal. So

Next, the trapezoidal sum is the average of the leftr sum and the right sum ,so

By elimination,

(b) Since the curve is concave up, the trapezoidal sum overestimates the integral.