trapezudemethode

The trapezoidal rule is a numerical method used to approximate the definite integral of a function. It is a simple yet effective technique that relies on approximating the area under a curve by dividing it into a series of trapezoids. The fundamental idea is to replace the curved function with straight line segments.

To apply the trapezoidal rule, the interval of integration is divided into a number of subintervals of equal width. Over each subinterval, the function's curve is approximated by a straight line connecting the function's values at the endpoints of the subinterval. This forms a trapezoid. The area of each trapezoid is then calculated using the formula for the area of a trapezoid: one-half the sum of the parallel sides multiplied by the height. In this context, the parallel sides are the function values at the subinterval endpoints, and the height is the width of the subinterval.

The total approximate integral is the sum of the areas of all these trapezoids. As the number of subintervals increases, the width of each subinterval decreases, and the approximation generally becomes more accurate. The trapezoidal rule is a foundational concept in numerical analysis and serves as a basis for more sophisticated integration techniques. It is particularly useful when the antiderivative of the function is difficult or impossible to find analytically.