trapezoid rule
The Trapezoid Rule is a method for approximating the value of a definite integral, characterized by its formula as an average of the Left and Right Riemann sums.
Conditions and Properties
The Trapezoid Rule and Midpoint Rule exhibit a consistent relationship in their error behavior for functions of consistent concavity, where the error of the Midpoint Rule is approximately half the magnitude and of opposite sign to the error of the Trapezoid Rule.
Applications
The Trapezoid Rule is a numerical method used to approximate the definite integral of a function over an interval. When the function is concave up on the interval, the Trapezoid Rule consistently overestimates the true value of the integral, whereas the Midpoint Rule underestimates it. Conversely, when the function is concave down, the Trapezoid Rule underestimates the integral, and the Midpoint Rule overestimates it.