Quiet@Remove["Global`*"];
SetDirectory[NotebookDirectory[]];
$HistoryLength=3;

expr[t_]:=a*(t^2)+b*t+c;
eqn={expr[tin-Δt]==fleft,
expr[tin]==fin,
expr[tin+Δt]==fright
};
solabc=Solve[eqn,{a,b,c}][[1]];
integrand=expr[t]/.solabc;
Integrate[integrand,{t,tin-Δt,tin+Δt}]
((%)/((Δt)/3))//Simplify

Quiet@Remove["Global`*"];
tmin=0.;
tmax=(Pi/2);
nIntervals=4;
Δt=((tmax-tmin)/nIntervals);
functionValues=Table[Sin[t],{t,tmin,tmax,Δt}];
weights={1,4,1};
triples=Partition[functionValues,3,2];
integral=((Δt)/3)Plus@@(triples/.{a_Real,b_,c_}:>weights.{a,b,c})

Quiet@Remove["Global`*"];
expr[t_]:=b*t+c;
eqn={expr[tleft]==fleft,
expr[tleft+Δt]==fright
};
solbc=Solve[eqn,{b,c}][[1]];
integrand=expr[t]/.solbc;
Integrate[integrand,{t,tleft,tleft+Δt}];

Quiet@Remove["Global`*"];
tmin=0.;
tmax=(Pi/2);
nIntervals=4;
Δt=((tmax-tmin)/nIntervals);
functionValues=Table[Sin[t],{t,tmin,tmax,Δt}];
weights={1,1};
triples=Partition[functionValues,2,1];
integral=((Δt)/2)Plus@@(triples/.{a_Real,b_}:>weights.{a,b})