trapetsoidisääntö

Trapetsoidisääntö, known in English as the trapezoidal rule, is a method for approximating the definite integral of a function. It is a simple yet effective numerical integration technique. The core idea behind the trapezoidal rule is to approximate the area under the curve of a function by dividing it into a series of trapezoids instead of rectangles, as used in methods like the midpoint rule or Riemann sums.

To apply the trapezoidal rule, the interval of integration is divided into several subintervals of equal width. Within each subinterval, the area under the function's curve is approximated by the area of a trapezoid whose parallel sides are the function's values at the endpoints of the subinterval and whose height is the width of the subinterval. The sum of the areas of all these trapezoids gives an approximation of the total definite integral.

The formula for the trapezoidal rule with n subintervals of width h = (b-a)/n, where [a, b] is the interval of integration, is given by:

∫[a to b] f(x) dx ≈ (h/2) * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x<n-1>) + f(x<n>)]

where xᵢ = a + i*h.

The accuracy of the trapezoidal rule generally increases as the number of subintervals (n) increases, leading to smaller trapezoids. While it is less accurate than more advanced methods like Simpson's rule for the same number of points, its simplicity makes it a useful tool for introductory numerical analysis and for approximating integrals when more complex methods are not required or feasible.