tailieunhanh - a guide to microsoft excel 2002 for scientists and engineers phần 8

Hội nhập số được sử dụng để đánh giá một tích xác định khi có biểu hiện không có hình thức đóng cửa cho tích phân hoặc khi các chức năng rõ ràng là không được biết đến và dữ liệu có sẵn chỉ ở dạng bảng. Số hội nhập (hoặc vuông góc) | 11 Numerical Integration Concepts We approximate the representative strip to a trapezoid. For a clearer drawing only five strips are used. Obviously more smaller strips are needed for a good approximation. Let there be n strips and hence n 1 data points. The area of a typical strip is A 1 X a AX Figure x b Numerical integration is used to evaluate a definite integral when there is no closed-form expression for the integral or when the explicit function is not known and the data is available in tabular form only. Numerical integration or quadrature consists of methods to find the approximate area under the graph of the function y x between two X values. The simplest of these uses the trapezoid rule. If we divide the area under the curve into a sufficiently large number of parts as shown in Figure then the area under the curve the approximate integral is given by . - 4 a 1 222 A Guide to Microsoft Excel 2002for Scientists and Engineers Ả òxy J-- 11-2 2 Combining the two equations we see that Ax Ax2 . Ax2ki2 IL3 2 2 2 Giving y v2y2 2j 2y4 --- 2y yn l 11-4 or Ax Í -- 2Ex . 1 2 Equation 1 is called the trapezoid rule. We use the trapezoid approximation in the first exercise to evaluate an integral. A better approximation to the integral is obtained by taking two adjacent strips and joining the three points on the curve with a parabola. This gives Equation called the Simpson one-third rule for approximating the area under a curve. 4 z 4Z 1 Z 2 Ax I 1 3 5 This rule requires that there be an even number of equally spaced strips. The second exercise uses this approximation. Approximating the curve through four adjacent points to a cubic equation gives the Simpson 3 arule shown in Equation . 1 X 3x 1 3z 2 z 3 Ar 7 Surprisingly the 3 a rule is often less accurate than the 1 3 rule. Elowever unlike the Vs rule it does not require an even number of strips. This is an advantage when the data is available only in tabular form the .