Chapter 7.7, Problem 24ES

Find a function $g(x)$ of the form

$g(x)=A{x}^2+Bx+C$

whose graph contains the points $\text{(m}-\Delta x,f(m-\Delta x)),(m,f(m)),$ and $(m+\Delta x,f(m+\Delta x)),$ for the given function $f(x)$ and the given values of $m$ and $\Delta x.$ Then verify Formula (11):

${\displaystyle {\int }_{m-\Delta x}^{m+\Delta x}g(x)dx=\frac{\Delta x}{3}[Y_0}+4Y_1+Y_2]$

$\text{where }{Y}_0=f(m-\Delta x),Y_1=f(m),\text{and}{Y}_2=f(m+\Delta x).$

$f(x)={\sin }^2(\pi x);m=\frac{1}{6},\Delta x=\frac{1}{6}$