puolisuunnikassäännössä

Puolisuunnikassääntö, or 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 the curve of a function by dividing it into a series of trapezoids.

The fundamental idea behind the puolisuunnikassääntö is to divide the interval of integration [a, b] into n subintervals of equal width, denoted by h. The width h is calculated as (b - a) / n. Within each subinterval, the function is approximated by a straight line connecting the function values at the endpoints of the subinterval. This creates a trapezoid.

The area of a single trapezoid in the i-th subinterval, from x_i to x_{i+1}, is given by the formula (f(x_i) + f(x_{i+1})) / 2 * h. To approximate the total integral, the areas of all these n trapezoids are summed up.

The formula for the puolisuunnikassääntö is:

Integral from a to b of f(x) dx ≈ h/2 * [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]

where x_0 = a and x_n = b.

As the number of subintervals, n, increases, the approximation generally becomes more accurate. The puolisuunnikassääntö is a foundational concept in numerical integration and serves as a basis for more advanced techniques. It is particularly useful when the antiderivative of the function is difficult or impossible to find analytically.