Simpson's Rule
This rule assumes a curve is a series of parabolas.
For a single parabola you have y(0) y(1) and y(2) at x(0), x(1), and x(2), equally spaced at distance h.
Area= 1/3*( y(0) + 4 * y(1) + y(2) )

Assume you have a curve with 6 equally divided panels.
x(0) to x(6) and y(0) to y(6)
Add results of each set of two panels to get total area.

Example

Assume f(x) = -3 * x^2 + 18

y(0)=18  y(1)=15  y(2)=6   y(3)=-9 y(4)=-30 y(5)=-57 y(6)=-90

Compute area of first panel pair.
y(0,2) = 1/3*(18+4*15+6) = 28

Compute area of second panel pair,
y(2,4) = 1/3*( 6+4*(-9)+(-30) ) = -20

Compute area of third panel pair,
y(4,6) = 1/3*( (-30)+4*(-57)+(-90) ) = -116

sum= area = -108

1/3 * (  ( y(0)=18)  +  ( y(6) = (-90) )   ) 
+ 2*(  ( y(2)=6 ) + ( y(4)=(-30)  ) +
4*( ( y(1)=15 ) + ( y(3)=(-9) ) + ( y(5)=-57 )   )
= -108

Integral {0 to 6} (-3 * x^2 + 18)  dx = -108
 
General rule
h= width of panel = 1 in this example
m= number of panels

Area=1/3 * h * (
( y(0) + y(2*m) ) + 
4 * ( y(1) + y(3) +...+ y(2*m-1) ) +
2 * ( y(2) + y(4) +...+ y(2*m-2) )  
)
--
Equation
y[1]=18   y[2]=15  y[3]=6  y[4]=-9  y[5]=-30 y[6]=-57  y[7]= -90
dim[y]=7->d
h=1

h*(y[1]+y[d]  +
2*sigma(y[n*2+1],n,1,(0.5*d-1.5) )   +
4*sigma(y[n*2]  ,n,1,(0.5*d-0.5) )
/3
->area

=====
Theory
x(0)=0  x(1)=1   x(2)=2
given y= a*x^2+b*x+c

a*(0^2)+b*0+c=y0
a*(1^2)+b*1+c=y1
a*(2^2)+b*2+c=y2

c=y0
a+b    =y1-y0
4*a+2*b=y2-y0

a=0.5*y0-y1+0.5*y2
b=-1.5*y0+2*y1-0.5*y2

Integrate a*x^2+b*x+c dx  { 0 to 2 }

area = 1/3*(y0+4*y1+y2)