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題名 | 定積分的近似值估計法=On Approximations to the Definite Integral |
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作者 | 陳榮治; Chen, Jung-chih; |
期刊 | 嘉義大學學報 |
出版日期 | 20000200 |
卷期 | 68 2000.02[民89.02] |
頁次 | 頁125-134 |
分類號 | 314.3 |
語文 | chi |
關鍵詞 | 定積分; 微積分基本定理; 黎曼和; 中點法; 梯形法; 拋物線法; 辛普森法; Definite integral; The fundamental theorem of calculus; Riemann sum; Midpoint rule; Trapezoidal rule; Parabolic rule; Simpson's rule; |
中文摘要 | 有些情況下,卻求一定積分□的精確值是不可能的,縱然我們已知f(x)是可積分的。但定積分的值存在是一回事,而求其值又是另一回事。 一種情況導因於當要利用微積分基本定理來估算此定積分□時,我們需要先尋f(x)的反導函數。不過,有時候欲尋求f(x)的反導函數,卻是非常困難,或甚至於根本不可能。另一種情況則起因於被積分式找不到適當的公式來配適。 這些情況下,我們不妨考慮求它的近似值。事實上;近似值的估計,在科學上的應用,常有一定的重要性,我們時常發現:有些目題真正精竘的解答太複雜,或根本無法得到,我們必須退而求其次,尋求一個較佳的近似解或近似值,它也常能顯示某種重要意義。 我們知道:定積分是定義為黎曼和的極限值,所以任意分割所得黎曼和之極限,均可視為近似積分值。此外,這裡我介紹了三種近似積分法,也就是中點法、梯形法,及拋物線法(或稱辛普森法)。再者,我們也針對每一種近似法,檢視其誤差之上界估計,並比較這三種方法之準確性。 當然,近似值的準確性,取決於函數的特性與分割之區間數n的大小,當n值愈大,所得近似值愈準確。 |
英文摘要 | There are some cases in which it is impossible to find the exact value of a definite integral. Although the definite integral □ must exist, evaluation is actually a very different mater. The first case arises from the fact that in order to evaluate □ suing the Fundamental Theorem of Calculus, we need to find antiderivative of f(x). However, sometimes it is really difficult or even impossible to find an antiderivative. Another case arises when three may be no formula for the integrand. In these cases, we need to find approximate value of definite integral. Besides the limit of any Riemann sum could be used as an approximation to the integral, here I present three useful methods: the Midpoint Rule, the Trapezoidal Rule and the Parabolic Rule. Furthermore, error bounds for each rule are also checked and compared. Of course, the accuracy of the approximation depends upon the nature of f(x) and the subintervals number n, it may be necessary to make n very large in order to obtain the desired degree of accuracy. |
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