Figure 6.7.8(a) shows a black curve with five points corresponding to $x\=x_k\,k\=0\,\dots \,4$, with the first three points interpolated with a quadratic whose graph is shown in red along the curve.

The rightmost of these three points, and the next two are interpolated by a second quadratic whose graph is shown in green along the curve.

These five nodes determine two contiguous "double panels." Simpson's rule amounts to approximating the area under a curve with the area under such approximating quadratics.

The line along which the two double panels in the figure touch deliberatelyshows both red and green.

Enter $\left[a\+\left(k\+n\right) h\,f_{k\+n}\right]$ and press the Enter key.

Context Panel: Sequence≻$n\=0\,2$

Context Panel: Conversions≻To List

Context Panel: Curve Fitting≻Polynomial Interpolation≻x

$\frac{1}{2}\frac{\left(f_{k\+2}-2f_{k\+1}\+f_k\right)x^2}{h^2}\+\frac{1}{2}\frac{\left(-2hk{f}_k\+4hk{f}_{k\+1}-2hk{f}_{k\+2}-2a{f}_k\+4a{f}_{k\+1}-2a{f}_{k\+2}-3h{f}_k\+4h{f}_{k\+1}-h{f}_{k\+2}\right)x}{h^2}\+\frac{1}{2}\frac{h^2{k}^2{f}_k-2h^2{k}^2{f}_{k\+1}\+h^2{k}^2{f}_{k\+2}\+2ahk{f}_k-4ahk{f}_{k\+1}\+2ahk{f}_{k\+2}\+3h^{2}k{f}_k-4h^{2}k{f}_{k\+1}\+h^{2}k{f}_{k\+2}\+a^2{f}_k-2a^2{f}_{k\+1}\+a^2{f}_{k\+2}\+3ah{f}_k-4ah{f}_{k\+1}\+ah{f}_{k\+2}\+2h^2{f}_k}{h^2}$

Context Panel: Definite Integral template
Integrate between $a\+k h$ and $a\+\left(k\+2\right) h$ then press the Enter key.

Context Panel: Simplify≻Simplify

$\frac{1}{3}h\left(4f_{k\+1}\+f_{k\+2}\+f_k\right)$

Expression palette: Evaluation template
Set $k\=0\,2\,4$ and add the results.

Context Panel: Factor

If $a\=x_0$ and $b\=x_n$, then there are $n\+1$ equispaced nodes in the interval $\left[a\,b\right]$. If $n$ is even, then the interval can be subdivided into $n\/2$ contiguous "double panels" spanning three nodes. These three nodes can be interpolated by a quadratic polynomial (whose graph is a parabola), and the area under the generic  parabola so formed is $A_k\=\frac{h}{3} \left(f_k\+4 f_{k\+1}\+f_{k\+2}\right)$. The sum of $n\/2$ areas of this type will then be