Topic 13. Differentiation and integration. Approximate methods of integration

1. Approximate methods of integration

Aditive quantities

The defined integral is the area of ​​the integral function. Then, the definition of an integral can be reduced to finding the sum of infinitely large numbers of infinitely small assembly sites.

The integral sum depends on two sets: 1) the choice of the value of h and 2) the choice of x on this interval according to the formula:

Formula of rectangles

Output: 1) the integral function f (x); 2) the interval of integration; 3) the number of integration intervals is predefined.

The essence of the method. For an approximate definition of the integral τf (x) dx, the segment [a; b] is divided into n equal parts by the points a = x0 <x1 <... <xn = b such that h = xi + 1-x = (ba) / n (i = 0.1, ..., n-1). Then the point x0 = a, x1 = a + h, xn = a + nh = b form an arithmetic program Maple S h a difference and are called nodes integration with integration step h. In these nodes, ordinates y0, y1, ..., yn are determined. The elementary segments [xi; xi + 1] find the square of the rectangle as the product f (xi) * h. The sum of such rectangles represents the approximate value of the integral.

If f (xi) is determined at the left ends of the segments [xi; xi + 1], then the formula of the left rectangles of the form is obtained:

                                       .

If f (xi) is defined in the right ends of the segments [xi; xi + 1], then the formula of the right rectangles of the form is obtained:

                                        .

If f (xi) are defined in the middle of the segments [xi; xi + 1], then the formula of the middle rectangles of the form is obtained:

                                .

Formula trapezoidal

Output data:

• integral function f (x);
• integration interval;
• The number of integration ranges is predefined.

The essence of the method. The formula of trapezys is obtained in the same way as the formula of rectangles. The only difference is that on each elementary segment [xi + 1-x] not a rectangle is constructed, but a trapezoid. Then the formula of the trapezium area will have the form:

                                         .

Simpson's formula

The Simpson formula is also obtained, as well as the formula of rectangles. The only difference is that on each elementary segment [xi + 1-x] not a rectangle is constructed, but a trapezoid. Then the Simpson formula will have the form:

                           .

Otherwise, this formula can be rewritten:

which is called the Simpson generalization formula.