Hello everybody,

I have a question as regards the TrapezoidalRule at the implemented NIntegrate function of Mathematica.

If I calculate the numeric(!) integral of f(x)=x^2 from 0 to 3 with 5 trapezoids by hand, I get a result of 9.18. If I use the implemented method "TrapezoidalRule" with the option "Points" -> 5 it gives me exactly 9.0.

So I doubt, that I do not fully understand what Mathematica does when using this Rule. I have tried a lot of (changing the WorkingPrecision, setting the MaxRecursions to 0 etc...), but it always comes up with the "real" (analytical) result.

Can anyone please explain what's going on?

Thanks a lot,

JJJ

f[x_] := x^2;
a = 0; b = 3;
\[CapitalDelta]x[n_] := (b - a)/n;
left[n_] := \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n - 1\)]\(f[
     a + i*\[CapitalDelta]x[n]]*\[CapitalDelta]x[n]\)\);
right[n_] := \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(f[
     a + i*\[CapitalDelta]x[n]]*\[CapitalDelta]x[n]\)\);
trap[n_] := (left[n] + right[n])/2;
trap2[n_] := \[CapitalDelta]x[n]/2 (f[a] + 2 \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n - 1\)]\(f[
        a + i*\[CapitalDelta]x[n]]\)\) + f[b]);
{left[5.], right[5.], trap[5.], trap2[5.]}
NIntegrate[f[x], {x, a, b}, 
 Method -> {"TrapezoidalRule", "Points" -> 3}, WorkingPrecision -> 10,
  MaxRecursion -> 0]