puolisuunnikassäännölle

The Puolisuunnikassääntö, or Trapezoidal Rule in English, is a numerical integration technique used to approximate the definite integral of a function. It's a fundamental method in numerical analysis for estimating the area under a curve when an analytical solution is difficult or impossible to obtain.

The core idea behind the Trapezoidal Rule is to divide the area under the curve of a function f(x) over an interval [a, b] into a series of smaller trapezoids. Instead of approximating the curve with a single straight line, as in the midpoint rule or rectangle rule, the Trapezoidal Rule uses a straight line connecting two points on the curve to form the top of each trapezoid.

To apply the rule, the interval [a, b] is divided into n subintervals of equal width, denoted by h = (b - a) / n. The endpoints of these subintervals are x0, x1, x2, ..., xn, where x0 = a and xn = b. The area of each trapezoid is calculated as the average of the function values at the endpoints of the subinterval, multiplied by the width of the subinterval. The total approximate integral is the sum of the areas of all these trapezoids.

The formula for the Trapezoidal Rule is given by:

Integral from a to b of f(x) dx ≈ (h/2) * [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)]

As the number of subintervals (n) increases, the approximation generally becomes more accurate. The Trapezoidal Rule provides a more accurate approximation than the rectangle rule because it takes into account the slope of the function within each subinterval.