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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.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
A right trapezoid (also called right-angled trapezoid) has two adjacent right angles. [13] Right trapezoids are used in the trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge. An obtuse trapezoid on the other hand has one acute and one obtuse angle on each base.
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six ...
Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the integral sign; Trigonometric substitution; Partial fractions in integration. Quadratic integral; Proof that 22/7 exceeds π; Trapezium rule; Integral of the secant function; Integral of secant cubed; Arclength; Solid of revolution ...
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.
The zeroeth extrapolation, R(n, 0), is equivalent to the trapezoidal rule with 2 n + 1 points; the first extrapolation, R(n, 1), is equivalent to Simpson's rule with 2 n + 1 points. The second extrapolation, R(n, 2), is equivalent to Boole's rule with 2 n + 1 points. The further extrapolations differ from Newton-Cotes formulas.
For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence. The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.