定積分的發(fā)展史

1、定積分的發(fā)展史 起源定積分的概念起源于求平面圖形的面積和其他一些實(shí)際問題。定積分的思想在古代數(shù)學(xué)家的工作中，就已經(jīng)有了萌芽。比如古希臘時期阿基米德在公元前240年左右，就曾用求和的方法計算過拋物線弓形及其他圖形的面積。公元 263 年我國劉徽提出的割圓術(shù)，也是同一思想。在歷史上，積分觀念的形成比微分要早。但是直到牛頓和萊布尼茨的工作出現(xiàn)之前（17世紀(jì)下半葉），有關(guān)定積分的種種結(jié)果還是孤立零散的，比較完整的定積分理論還未能形成，直到牛頓-萊布尼茨公式建立以后，計算問題得以解決，定積分才迅速建立發(fā)展起來。The next significant advances in integral calcu

2、lus did not begin to appear until the 16th century.未來的重大進(jìn)展，在微積分才開始出現(xiàn)，直到16世紀(jì)。 At this time the work of Cavalieri with his method of indivisibles , and work by Fermat , began to lay the foundations of modern calculus, with Cavalieri computing the integrals of x n up to degree n = 9 in Cavalieris quadr

3、ature formula .此時的卡瓦列利與他的indivisibles方法 ，并通過費(fèi)爾馬工作，開始卡瓦列利計算度N = 9 N的積分奠定現(xiàn)代微積分的基礎(chǔ)， 卡瓦列利的正交公式 。Further steps were made in the early 17th century by Barrow and Torricelli , who provided the first hints of a connection between integration and differentiation 17世紀(jì)初Barrow provided the first proof of the fu

4、ndamental theorem of calculus . Wallis generalized Cavalieris method, computing integrals of x to a general power, including negative powers and fractional powers.巴羅提供的第一個證明微積分基本定理。 At around the same time, there was also a great deal of work being done by Japanese mathematicians , particularly by S

5、eki Kwa . 3 He made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of exhaustion . edit Newton and Leibniz牛頓和萊布尼茨 The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of

6、 calculus by Newton and Leibniz .在一體化的重大進(jìn)展是在17世紀(jì)獨(dú)立發(fā)現(xiàn)的牛頓 和 萊布尼茨的微積分基本定理。 The theorem demonstrates a connection between integration and differentiation.定理演示了一個整合和分化之間的連接。 This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals.這方面，分化比較容易地結(jié)合起來，可以利

7、用來計算積分。 In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems.特別是微積分基本定理，允許一個要解決的問題更廣泛的類。 Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed.同等重要的是，牛頓和萊布尼茨開發(fā)全面的數(shù)學(xué)框架。 Given the name infinitesimal cal

8、culus, it allowed for precise analysis of functions within continuous domains.由于名稱的微積分，它允許精確的分析在連續(xù)域的功能。 This framework eventually became modern calculus , whose notation for integrals is drawn directly from the work of Leibniz.這個框架最終成為現(xiàn)代微積分符號積分是直接從萊布尼茨的工作。 edit Formalizing integrals正式積分 定積分概念的理論基礎(chǔ)是極

9、限。人類得到比較明晰的極限概念，花了大約2000年的時間。在牛頓和萊布尼茨的時代，極限概念仍不明確。因此牛頓和萊布尼茨建立的微積分的理論基礎(chǔ)還不十分牢靠，有些概念還比較模糊，由此引起了數(shù)學(xué)界甚至哲學(xué)界長達(dá)一個半世紀(jì)的爭論，并引發(fā)了“第二次數(shù)學(xué)危機(jī)”。經(jīng)過十八、十九世紀(jì)一大批數(shù)學(xué)家的努力，特別是柯西首先成功地建立了極限理論，魏爾斯特拉斯進(jìn)一步給出了現(xiàn)在通用的極限的 定義，極限概念才完全確立，微積分才有了堅實(shí)的基礎(chǔ)，也才有了我們今天在教材中所見到的微積分?，F(xiàn)代教科書中有關(guān)定積分的定義是由黎曼給出的。 edit Terminology and notation術(shù)語和符號 Isaac Newton u

10、sed a small vertical bar above a variable to indicate integration, or placed the variable inside a box. 艾薩克牛頓以上的變量使用一個小豎線表示一體化，或放置在一個盒子里的變量， The vertical bar was easily confused with豎線是很容易混淆。 or或 , which Newton used to indicate differentiation, and the box notation was difficult for printers to repr

11、oduce, so these notations were not widely adopted.牛頓用來指示分化和方塊符號打印機(jī)難以重現(xiàn)，所以這些符號沒有被廣泛采用。 The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675 ( Burton 1988 , p. 359; Leibniz 1899 , p. 154).1675 年戈特弗里德萊布尼茨He adapted the integral symbol , , from the letter ( long s

12、), standing for summa (written as umma ; Latin for sum or tot改編的積分符號 ，從字母 S（“總結(jié)”或“總”）。 The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mmoires of the French Academy around 181920, reprinted in his book of 1822 ( Cajori

13、 1929 , pp. 249250; Fourier 1822 , 231).The sign represents integration; a and b are the lower limit and upper limit , respectively, of integration, defining the domain of integration; f is the integrand, to be evaluated as x varies over the interval a , b ; and dx is the variable of integration . 符

14、號表示的整合; A和 B 的下限和上限 ，分別一體化，定義域的融合; f是積，x在區(qū)間a，b上的變化進(jìn)行評估; Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit of weighted sums, so that the dx suggested the limit of a difference (namely, the interval width).從歷史上看，黎曼嚴(yán)格解

15、釋無窮小的早期努力失敗后，正式定義為積分的加權(quán)求和的限制， 使有差別的限制（即間隔寬度）。 Shortcomings of Riemanns dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral , which is founded on an ability to extend the idea of measure in much more flexible ways.黎曼的間隔和連續(xù)性的依賴的缺點(diǎn)促使了新的定義，尤其是勒貝格積分，這是建立能力

16、，延長了“措施”，以更靈活的方式的想法。 Thus the notation因此，符號 refers to a weighted sum in which the function values are partitioned, with measuring the weight to be assigned to each value.是指在分區(qū)函數(shù)值測量的重量被分配到每個值，加權(quán)總和。 Here A denotes the region of integration.在這里，A表示一體化的地區(qū)。 定積分既是一個基本概念，又是一種基本思想。定積分的思想即“化整為零近似代替積零為整取極限”。定積分這種“和的極限”的思想，在高等數(shù)學(xué)、物理、工程技術(shù)、其他的知識領(lǐng)