定积分的发展史

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 Cavalieri's 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 Cavalieri's 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。金属活接 At around the same time， there was also a great deal of work being done by Japanese mathematicians , particularly by Seki Kōwa 。 [ 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。 通用积分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.信息包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

［ 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 ſ （ long s ）, standing for summa (written as ſumma ； 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 Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 ( Cajori 1929 , pp. 249–250； Fourier 1822 ， §231).

The ∫ sign represents integration； a清洗空调现摄像头 and b are the CL在性方面是什么意思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 Riemann's 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