Chapter 7.7, Problem 54ES

Suppose that $f$ is a continuous nonnegative function on the intervals [a, b], $n$ is even, and [a, b] is partitioned using $n+1$ equally spaced points, $a=x_0<x_1<\mathrm{...}<x_n=b.{\text{Set y}}_0=f(x_0),y_1=f(x_1),\mathrm{...},y_n=f(x_n).$ Let $g_1,g2,\mathrm{...},g_n/2$ be the quadratic functions of the form $g_i(x)=A{x}^2+Bx+C\text{so}\text{that}$

$•$ the graph of $g_1$ passes through the points $(x_0,y_0),(x_1,y_1),\text{and}\text{(}{x}_2,y_2);$

$•$ the graph of $g_2$ passes through the points $(x_2,y_2),(x_3,y_3),$ $(x_4,y_4);$

$•$ …

$•$ the graph of $g_{n/2}$ passes through the points $(x_{n-2},y_{n-2}),(x_{n-1},y_{n-1}),\text{and}(x_n,y_n).$

Verify that Formula (8) computes the area under a piecewise quadratic function by showing that

${\displaystyle \underset{j=1}{\overset{n/2}{\sum }}\left({\displaystyle {\int }_{x_{2j-2}}^{x_{2j}}{g}_j(x)dx}\right)}=\frac{1}{3}\left(\frac{b-a}{n}\right)[y_0+4y_1+2y_2+4y_3+2y_4+\mathrm{...}+2y_{n-}{}_2+4y_{n-1}+y_n]$