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In calculus, the trapezoidal rule is a technique for approximating the definite integral.
The trapezoidal rule works by approximating the region under the graph of the function f {\displaystyle f} as a trapezoid and calculating its area. It follows that
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule". Let { x k } {\displaystyle \{x_{k}\}} be a partition of {\displaystyle } such that a = x 0 < x 1 < ⋯ < x N − 1 < x N = b {\displaystyle a=x_{0} When the partition has a regular spacing, as is often the case, that is, when all the Δ x k {\displaystyle \Delta x_{k}} have the same value Δ x , {\displaystyle \Delta x,} the formula can be simplified for calculation efficiency by factoring Δ x {\displaystyle \Delta x} out:.