trapezregeln

Trapezregeln, also known as the trapezoidal rule, are a family of numerical methods used for approximating the definite integral of a function. The fundamental idea behind trapezoidal rules is to approximate the area under the curve of a function by dividing the integration interval into smaller subintervals and approximating the area within each subinterval using a trapezoid.

The simplest form is the single trapezoidal rule. Given a function f(x) and an interval [a, b], the area is approximated by the area of a single trapezoid whose parallel sides are the function values at the endpoints, f(a) and f(b), and whose height is the width of the interval, b - a. The formula is (b - a) * [f(a) + f(b)] / 2.

More accurate approximations are achieved using the composite trapezoidal rule. This involves dividing the interval [a, b] into n equal subintervals of width h = (b - a) / n. The area is then approximated by summing the areas of n trapezoids, each with width h and heights corresponding to the function values at the endpoints of each subinterval. The formula for the composite trapezoidal rule is h/2 * [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)], where xi = a + i*h.

Trapezoidal rules are a basic but important technique in numerical integration and serve as a foundation for more advanced methods. Their accuracy generally increases with the number of subintervals used.