Chapter 7.7, Problem 51ES

(a) Verify that the average of the left and right end-point approximations as given in Table 7.7.1 gives Formula (2) for the trapezoidal approximation.

(b) Suppose that $f$ is a continuous nonnegative function on the interval [a, b] and partition [a, b] with equally spaced points, $a=x_0<x_1<\mathrm{...}<x_n=b.$ Find the area of the trapezoid under the line segment joining points $(x_k,f(x_k))\text{and}\text{(}{x}_{k+1},f(x_{k+1}))$ and above the interval $\left[x_k,x_{x_{k+1}}\right].$ Show that the right side of Formula (2) is the sum of these trapezoidal areas (Figure 7.7.1).