Trapezoidal rule

In calculus, the trapezoidal rule (informally trapezoid rule; or in British English trapezium rule) is a technique for numerical integration, i.e. approximating the definite integral: ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.} 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. This is easily calculated by noting that the area of the region is made up of a rectangle with width ( b − a ) {\displaystyle (b-a)} and height f ( a ) {\displaystyle f(a)} , and a triangle of width ( b − a ) {\displaystyle (b-a)} and height f ( b ) − f ( a ) {\displaystyle f(b)-f(a)} . Therefore, ∫ a b f ( x ) d x ≈ ( b − a ) ⋅ f ( a ) ⏟ Area of rectangle + 1 2 ( b − a ) ⋅ [ f ( b ) − f ( a ) ] ⏟ Area of triangle = ( b − a ) ⋅ ( f ( a ) + 1 2 f ( b ) − 1 2 f ( a ) ) = ( b − a ) ⋅ ( 1 2 f ( a ) + 1 2 f ( b ) ) = 1 2 ( b − a ) [ f ( a ) + f ( b ) ] . {\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx \underbrace {(b-a)\cdot f(a)} _{\text{Area of rectangle}}+\underbrace {{\tfrac {1}{2}}(b-a)\cdot [f(b)-f(a)]} _{\text{Area of triangle}}\\&=(b-a)\cdot \left(f(a)+{\tfrac {1}{2}}f(b)-{\tfrac {1}{2}}f(a)\right)\\&=(b-a)\cdot \left({\tfrac {1}{2}}f(a)+{\tfrac {1}{2}}f(b)\right)\\&={\frac {1}{2}}(b-a)[f(a)+f(b)].\end{aligned}}} The rule can also be derived by replacing the integrand with the equation of the line joining points ( a , f ( a ) ) {\displaystyle {\big (}a,f(a){\big )}} and ( b...