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课程名称：MATH 166：微积分 II - AUM 课程概述： 本课程涵盖微积分的深入内容，主要集中在积分的概念及其应用。 章节5.1介绍了曲线下的面积和积分的基本概念，说明如何使用定积分通过对函数图下无穷小矩形的求和来计算面积，为理解积分与量的累积之间的关系奠定基础。 章节5.3讨论了微积分基本定理（FTC），该定理连接了微分与积分，表明如果一个函数在某个区间内是连续的，那么它的导数的定积分等于函数值的净变化。这一理论简化了使用反导数计算定积分的过程。 章节5.4探讨了不定积分，表示反导数的家族，引入了任意常数的概念，因为不定积分没有特定的边界。本章包括对幂函数、指数函数和三角函数的积分规则。 章节5.5介绍了代换法，这是一种通过逆转微分链式法则来评估积分的强大技术，尤其适用于处理复合函数以及与已知函数的导数相似的积分。 章节7.1介绍了分部积分法，这种技术基于乘积规则，适用于函数乘积的积分，例如xex和xlnx等。 章节7.2涵盖三角积分，专注于正弦、余弦、割线和正切函数的积分，并通过各种三角恒等式来简化这些积分。 章节7.3讨论了三角替换法，这是一种通过代入三角表达式（如x=sinθ）来简化复杂根式的积分方法。 章节7.4解释了使用部分分式法对有理函数的积分，这种技术将分式分解为更简单的项，从而更易于积分。 章节7.8介绍了不当积分，涉及无限界限或无界函数，通过极限分析收敛性与发散性。 章节6.1讲解了通过对两个函数在一个区间内的差积分来计算曲线之间的面积，这一技术在物理和工程应用中非常有用。 这些主题构成了积分微积分的基础及其在各个领域的应用。
Course contentCH 5.1 Areas and Distances CH 5.3 The Fundamental Theorem of CalculusCH 5.4 Indefinite Integrals CH 5.5 The Substitution RuleCH 7.1 Integration by PartsCH 7.2 Trigonometric IntegralsCH 7.3 Trigonometric SubstitutionCH 7.4 Integration of Rational Functions by Partial FractionsCH 7.8 improper integralCH 6.1 Areas Between Curves Chapter 5.1 introduces the concept of areas under curves and the fundamental idea of integration. It explains how definite integrals can be used to compute areas by summing infinitely small rectangles under a function's graph. This section lays the groundwork for understanding the relationship between integration and accumulation of quantities.Chapter 5.3 discusses the Fundamental Theorem of Calculus (FTC), which connects differentiation and integration. The FTC states that if a function is continuous over an interval, the definite integral of its derivative gives the net change in the function's values. This theorem simplifies the computation of definite integrals using antiderivatives.Chapter 5.4 explores indefinite integrals, which represent families of antiderivatives. The concept of an arbitrary constant, CCC, is introduced, as indefinite integrals lack specific boundaries. This chapter includes rules for integrating power, exponential, and trigonometric functions.Chapter 5.5 presents the substitution rule, a powerful technique for evaluating integrals by reversing the chain rule of differentiation. This method is especially useful when dealing with composite functions and integrals that resemble derivatives of known functions.Chapter 7.1 introduces integration by parts, based on the product rule of differentiation. This technique is essential for integrating products of functions, such as xexx e^xxex or xlnxx /ln xxlnx.Chapter 7.2 covers trigonometric integrals, focusing on integrals involving sine, cosine, secant, and tangent functions. Various trigonometric identities simplify these integrals.Chapter 7.3 discusses trigonometric substitution, a method for integrating functions involving square roots by substituting trigonometric expressions, such as x=sinθx = /sin /thetax=sinθ, to simplify complex radicals.Chapter 7.4 explains the integration of rational functions using partial fractions. This technique decomposes fractions into simpler terms, making them easier to integrate.Chapter 7.8 introduces improper integrals, which involve infinite limits or unbounded functions. Convergence and divergence are analyzed using limits.Chapter 6.1 covers areas between curves by integrating the difference of two functions over an interval. This technique is useful in physics and engineering applications.These topics form the foundation of integral calculus and its applications in various fields.
