Editing Riemann sum (section)

=== Trapezoidal rule ===
{{main|Trapezoidal rule}}
[[Image:TrapRiemann2.svg|thumb|right|Trapezoidal sum of {{math|''x'' ↦ ''x''<sup>3</sup>}} over {{math|[0, 2]}} using 4 subintervals]]
For the trapezoidal rule, the function is approximated by the average of its values at the left and right endpoints of the subintervals. Using the area formula <math>\tfrac{1}{2}h(b_1 + b_2)</math> for a [[trapezoid|trapezium]] with parallel sides {{math|''b''<sub>1</sub>}} and {{math|''b''<sub>2</sub>}}, and height {{mvar|h}}, and summing the resulting areas gives
<math display="block">S_\mathrm{trap} = \tfrac{1}{2}\Delta x\left[f(a) + 2f(a + \Delta x) + 2f(a + 2\Delta x) + \dots + f(b)\right].</math>

The error of this formula will be
<math display="block">\left\vert\int_a^b f(x)\, dx - S_\mathrm{trap}\right\vert \le \frac{M_2(b - a)^3}{12n^2},</math>
where <math>M_2</math> is the maximum value of the absolute value of <math>f''(x)</math>.

The approximation obtained with the trapezoidal sum for a function is the same as the average of the left hand and right hand sums of that function.