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The midpoint rule uses the function value at the centre of each strip, which cancels the linear part of the error. It is typically about twice as accurate as the trapezoid rule and errs in the opposite direction.
Midpoint rule
∫ ≈ h·Σ f(a + (k + ½)h)
The exact value is 9 and the midpoint rule gives 8.9375, an error of 0.0625.
The trapezoid rule's error is about twice as large and of opposite sign; averaging the two in a 2:1 ratio gives Simpson's rule.