定积分的发展史

定积分的发展史 起源定积分的概念起源于求平面图形的面积和其他一些实际问题。定积分的思想在古代数学家的工作中，就已经有了萌芽。比如古希腊时期阿基米德在公元前240年左右，就曾用求和的方法计算过抛物线弓形及其他图形的面积。公元 263 年我国刘徽提出的割圆术，也是同一思想。在历史上，积分观念的形成比微分要早。但是直到牛顿和莱布尼茨的工作出现之前（17世纪下半叶），有关定积分的种种结果还是孤立零散的，比较完整的定积分理论还未能形成，直到牛顿-莱布尼茨公式建立以后，计算问题得以解决，定积分才迅速建立发展起来。The next significant advances in integral calculus did not begin to appear until the 16th century.未来的重大进展，在微积分才开始出现，直到16世纪。 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 quadrature formula .此时的卡瓦列利与他的indivisibles方法 ，并通过费尔马工作，开始卡瓦列利计算度N = 9 N的积分奠定现代微积分的基础， 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世纪初Barrow provided the first proof of the fundamental theorem of calculus . Wallis generalized Cavalieris method, computing integrals of x to a general power, including negative powers and fractional powers.巴罗提供的第一个证明微积分基本定理。 At around the same time, there was also a great deal of work being done by Japanese mathematicians , particularly by Seki 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牛顿和莱布尼茨 The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz .在一体化的重大进展是在17世纪独立发现的牛顿 和 莱布尼茨的微积分基本定理。 The theorem demonstrates a connection between integration and differentiation.定理演示了一个整合和分化之间的连接。 This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals.这方面，分化比较容易地结合起来，可以利用来计算积分。 In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems.特别是微积分基本定理，允许一个要解决的问题更广泛的类。 Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed.同等重要的是，牛顿和莱布尼茨开发全面的数学框架。 Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains.由于名称的微积分，它允许精确的分析在连续域的功能。 This framework eventually became modern calculus , whose notation for integrals is drawn directly from the work of Leibniz.这个框架最终成为现代微积分符号积分是直接从莱布尼茨的工作。 edit Formalizing integrals正式积分 定积分概念的理论基础是极限。人类得到比较明晰的极限概念，花了大约2000年的时间。在牛顿和莱布尼茨的时代，极限概念仍不明确。因此牛顿和莱布尼茨建立的微积分的理论基础还不十分牢靠，有些概念还比较模糊，由此引起了数学界甚至哲学界长达一个半世纪的争论，并引发了“第二次数学危机”。经过十八、十九世纪一大批数学家的努力，特别是柯西首先成功地建立了极限理论，魏尔斯特拉斯进一步给出了现在通用的极限的 定义，极限概念才完全确立，微积分才有了坚实的基础，也才有了我们今天在教材中所见到的微积分。现代教科书中有关定积分的定义是由黎曼给出的。 edit Terminology and notation术语和符号 Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. 艾萨克牛顿以上的变量使用一个小竖线表示一体化，或放置在一个盒子里的变量， The vertical bar was easily confused with竖线是很容易混淆。 or或 , which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.牛顿用来指示分化和方块符号打印机难以重现，所以这些符号没有被广泛采用。 The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675 ( Burton 1988 , p. 359; Leibniz 1899 , p. 154).1675 年戈特弗里德莱布尼茨He adapted the integral symbol , , from the letter ( long s ), standing for summa (written as umma ; Latin for sum or tot改编的积分符号 ，从字母 S（“总结”或“总”）。 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 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 . 符号表示的整合; A和 B 的下限和上限 ，分别一体化，定义域的融合; f是积，x在区间a，b上的变化进行评估; 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).从历史上看，黎曼严格解释无穷小的早期努力失败后，正式定义为积分的加权求和的限制， 使有差别的限制（即间隔宽度）。 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.黎曼的间隔和连续性的依赖的缺点促使了新的定义，尤其是勒贝格积分，这是建立能力，延长了“措施”，以更灵活的方式的想法。 Thus the notation因此，符号 refers to a weighted sum in which the function values are partitioned, with measuring the weight to be assigned to each value.是指在分区函数值测量的重量被分配到每个值，加权总和。 Here A denotes the region of integration.在这里，A表示一体化的地区。 定积分既是一个基本概念，又是一种基本思想。定积分的思想即“化整为零近似代替积零为整取极限”。定积分这种“和的极限”的思想，在高等数学、物理、工程技术、其他的知识领