Example 2:  f[x]=Sin[x] 

f[x_]:=Sin[x];  a=0; b=Pi/2;
FilledPlot[f[x],{x,a,b}]

[Graphics:../Images/simpson_gr_24.gif]

INT=Integrate[f[x],{x,a,b}]; {INT,N[INT]}
[Graphics:../Images/simpson_gr_25.gif]

n=1 

n=1;
LEFT[n]=LeftEndpointRule[f[x],{x,a,b,n}]; {LEFT[n],N[LEFT[n]]}
[Graphics:../Images/simpson_gr_26.gif]
RIGHT[n]=RightEndpointRule[f[x],{x,a,b,n}];  {RIGHT[n],N[RIGHT[n]]}
[Graphics:../Images/simpson_gr_27.gif]
TRAP[n]=TrapezoidRule[f[x],{x,a,b,n}];  {TRAP[n],N[TRAP[n]]}
[Graphics:../Images/simpson_gr_28.gif]
MID[n]=MidpointRule[f[x],{x,a,b,n}];  {MID[n],N[MID[n]]}
[Graphics:../Images/simpson_gr_29.gif]
SIMP[n]=SimpsonRule[f[x],{x,a,b,n}];  {SIMP[n],N[SIMP[n]]}
[Graphics:../Images/simpson_gr_30.gif]

n=2 

n=2;
LEFT[n]=LeftEndpointRule[f[x],{x,a,b,n}]; {LEFT[n],N[LEFT[n]]}
[Graphics:../Images/simpson_gr_31.gif]
RIGHT[n]=RightEndpointRule[f[x],{x,a,b,n}];  {RIGHT[n],N[RIGHT[n]]}
[Graphics:../Images/simpson_gr_32.gif]
TRAP[n]=TrapezoidRule[f[x],{x,a,b,n}];  {TRAP[n],N[TRAP[n]]}
[Graphics:../Images/simpson_gr_33.gif]
MID[n]=MidpointRule[f[x],{x,a,b,n}];  {MID[n],N[MID[n]]}
[Graphics:../Images/simpson_gr_34.gif]
SIMP[n]=SimpsonRule[f[x],{x,a,b,n}];  {SIMP[n],N[SIMP[n]]}
[Graphics:../Images/simpson_gr_35.gif]

n=3 

n=3;
LEFT[n]=LeftEndpointRule[f[x],{x,a,b,n}]; {LEFT[n],N[LEFT[n]]}
[Graphics:../Images/simpson_gr_36.gif]
RIGHT[n]=RightEndpointRule[f[x],{x,a,b,n}];  {RIGHT[n],N[RIGHT[n]]}
[Graphics:../Images/simpson_gr_37.gif]
TRAP[n]=TrapezoidRule[f[x],{x,a,b,n}];  {TRAP[n],N[TRAP[n]]}
[Graphics:../Images/simpson_gr_38.gif]
MID[n]=MidpointRule[f[x],{x,a,b,n}];  {MID[n],N[MID[n]]}
[Graphics:../Images/simpson_gr_39.gif]
SIMP[n]=SimpsonRule[f[x],{x,a,b,n}];  {SIMP[n],N[SIMP[n]]}
[Graphics:../Images/simpson_gr_40.gif]

n=4 

n=4;
LEFT[n]=LeftEndpointRule[f[x],{x,a,b,n}]; {LEFT[n],N[LEFT[n]]}
[Graphics:../Images/simpson_gr_41.gif]
RIGHT[n]=RightEndpointRule[f[x],{x,a,b,n}];  {RIGHT[n],N[RIGHT[n]]}
[Graphics:../Images/simpson_gr_42.gif]
TRAP[n]=TrapezoidRule[f[x],{x,a,b,n}];  {TRAP[n],N[TRAP[n]]}
[Graphics:../Images/simpson_gr_43.gif]
MID[n]=MidpointRule[f[x],{x,a,b,n}];  {MID[n],N[MID[n]]}
[Graphics:../Images/simpson_gr_44.gif]
SIMP[n]=SimpsonRule[f[x],{x,a,b,n}];  {SIMP[n],N[SIMP[n]]}
[Graphics:../Images/simpson_gr_45.gif]

Simpson's rule with n=1..20, as compared to exact answer 

ColumnForm[Table[N[SimpsonRule[f[x],{x,a,b,n}],20],{n,20}]]
1.002279877492210477707867113605819
1.000134584974193904475917036123415
1.000026312170592824146325550965273
1.000008295523967758783487861510767
1.000003392220900552095136890838824
1.000001634438579846666066431472377
1.000000881751272127448526674740433
1.000000516684706500345535322980275
1.000000322485988642732731117203464
1.0000002115465914236020780056877
1.000000144470745199500486976271696
1.000000101996096953633722821594001
1.000000074046128068239124191472887
1.000000055047502057421420912568325
1.000000041769929766375779822275749
1.000000032265000961551117744244116
1.000000025316385447025035327927104
1.000000020141668377900460878534619
1.000000016224113187661863333999933
1.000000013214379464198167439030713 
N[INT,20]
[Graphics:../Images/simpson_gr_46.gif]

Converted by Mathematica      February 18, 2001