trapezoidregeln

The trapezoidal rule is a numerical method used to approximate the definite integral of a function. It is a basic technique in numerical analysis and is based on approximating the area under the curve of a function by dividing it into several trapezoids. Instead of approximating the area with rectangles as in the simpler rectangular rule, the trapezoidal rule uses trapezoids, which generally leads to a more accurate approximation.

To apply the trapezoidal rule, the interval of integration is divided into a number of subintervals of equal width. On each subinterval, the function is approximated by a straight line connecting the function values at the endpoints of the subinterval. This creates a trapezoid whose area can be calculated. The sum of the areas of all these trapezoids then approximates the total area under the curve, and thus the value of the definite integral. The formula for the trapezoidal rule involves summing the average of the function values at the endpoints of each subinterval, multiplied by the width of the subinterval. As the number of subintervals increases, the approximation generally becomes more accurate. The trapezoidal rule is a fundamental method for numerical integration and serves as a building block for more sophisticated quadrature rules.