Chapter 8.8, Problem 62E

To determine

To show that if $f(x)=p{x}^2+qx+r$ is a quadratic polynomial, then $S_2={\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}$. In other words, show that ${\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}=\frac{b-a}{6}(y_0+4y_1+y_2)$ where $y_0=f(a)$, $y_1=f(\frac{a+b}{2})$ and $y_2=f(b)$ using the hint that we need to show this first for $f(x)=1,x,x^2$ and use linearity