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Euler method and midpoint method, related methods for solving differential equations; Lebesgue integration; Riemann integral, limit of Riemann sums as the partition becomes infinitely fine; Simpson's rule, a powerful numerical method more powerful than basic Riemann sums or even the Trapezoidal rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) [a] is a technique for numerical integration, i.e., approximating the definite integral: (). The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area.
The midpoint method computes + so that the red chord is approximately parallel to the tangent line at the midpoint (the green line). In numerical analysis , a branch of applied mathematics , the midpoint method is a one-step method for numerically solving the differential equation ,
Suppose that we want to solve the differential equation ′ = (,). The trapezoidal rule is given by the formula + = + ((,) + (+, +)), where = + is the step size. [1]This is an implicit method: the value + appears on both sides of the equation, and to actually calculate it, we have to solve an equation which will usually be nonlinear.
After trapezoid rule estimates are obtained, Richardson extrapolation is applied. For the first iteration the two piece and one piece estimates are used in the formula 4 × (more accurate) − (less accurate) / 3 . The same formula is then used to compare the four piece and the two piece estimate, and likewise for the higher estimates
The ancient Babylonians used the trapezoidal rule to integrate the motion of Jupiter along the ecliptic. [3] Antique method to find the Geometric mean. For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side = (the Geometric mean of a and b).