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双语推荐：积分学

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积分学是高等数学的主要分支。定积分与二重积分的概念和计算是积分学的重要组成部分，其在在几何、物理、概率统计等方面有着重要应用。二重积分作为定积分的推广，其二者有着共同的解题技巧，包括利用几何意义、对称性、换元法来简化定积分与二重积分的计算。
Calculus is the main branch of higher mathematics. Concepts and calculations of Definite integral and double integral are important parts of the calculus, which has important applications in geometry, physics, Probability and Statistics and so on. Double integral as the promotion of definite integrals, which both share a common problem-solving skills, including the use of the geometric sense, symmetry, changing element method to simplify the calculation of definite integrals and double integrals.

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积分不等式是数学分析的一个重要内容.针对曲线积分和曲面积分不等式问题,本文利用条件极值及曲线积分和曲面积分的性质建立几个不等式,并给予证明,旨在培养学生的创新和发散思维能力,也为教师在教学过程中提供一些思想方法.
Integral inequality is an important element in mathematical analysis teaching .Some curvilinear integral and surface integral inequalities are structured and proven using conditional extremum and properties of curvilinear integral and surface integral in this paper .The aim is to cultivate students''innovation ability and di-vergent thinking ability , but also offer some ways of thinking in teacher''s teaching process .

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由参数方程表示的曲线在积分学中占有重要地位，它和定积分的换元积分法的密切联系。在求由参数曲线所围平面图形面积时，积分限的确定是一个难点，它与参数曲线的多重点息息相关。
The curve shown by the parameter equation, which is closely bound up with integration by substitu-tion of definite integral, plays an important part in integral calculus. When calculating the area of plane en-closed by the parameter curve,one may find it is a difficult point to fix the integral bounds, which is vitally in-terrelated with multiple repeating point of parameter curve. This paper concerns a superficial understanding of multiple repeating point of parameter curve.

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不定积分是高等数学积分学中的重要组成部分。不定积分的计算方法比较灵活，学生在学习这部分计算时感觉比较困难。本文通过例题介绍其常用的一些方法，使学生快速掌握求解技巧，较好地掌握这部分知识，提高解题能力，为定积分的计算奠定基础。
Indefinite integral is an important part of integral cal-culus in"Higher Mathematics". The computation methods of in-definite integral are relatively flexible, making students feel diffi-cult in the computation. Through a case-based introduction of some common methods, this paper aims to help students rapidly grasp solving skills, better master the knowledge and improve the ability of problem solving, thus laying a foundation for the com-putation of definite integral.

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阐述并举例说明了两种对称性在多元函数积分学中的应用.
Two types of symmetries are summarized and illustrated for the calculation of integrals of multivariable functions .

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在实际问题和数学分析后续课程(如概率论)中,经常出现广义Riemann积分。但是我们发现,现有教科书上对此类积分的研究都是基于定积分的思想方法,要求被积函数有一定的光滑性,这大大限制了广义积分的研究范围。该文研究Lebesgue积分方法在广义Riemann积分的收敛性判别和计算以及含参量广义Riemann积分性质等问题中的应用。通过理论与实例结合,充分说明了Lebesgue方法的简便与灵活。因此,我们在学习广义Riemann积分时,不应拘泥于教科书上的现有知识和方法,应该拓宽思路,合理结合其他的课程。
Generalized Riemann integral is very useful in practical problems and subsequent courses of mathematical analysis(such as Probability theory).However,the study of such points in our textbooks is based on the methods of definite integral,requiring the functions with certain smoothness,which greatly limits the research scope of generalized integral.In this paper,we discuss some applications of Lebesgue methods in generalized Riemann integral about convergence and calculation as well as generalized Riemann integral with parameters.Combined theory with examples,we show the simplicity and flexibility of Lebesgue methods.When we study the generalized Riemann integral,therefore,should not be constrained by the existing knowledge and method of textbook.We should broaden the train of thought and combine with other courses appropriately.

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光学遥感卫星中空间相机常用的离轴三反光学系统，离轴三反系统能扩大光学系统视场、提高系统调制传递函数（MTF），但是离轴角会带来积分时间不同步的问题，文章对离轴角引起的积分时间不同步而产生的成像品质影响进行了分析。首先对离轴三反光学系统的离轴角建立数学模型后，应用STK软件仿真空间TDICCD相机在不同级数下离轴角带来的摄影点积分时间和星下点积分时间的差异，然后用Matlab编写程序处理得到的不同积分时间的采样点得出结论，离轴角越大，光学系统的传递函数越小，光学相机成像品质越差。在工程应用中，对一个确定离轴角的离轴三反光学系统，通过仿真，对积分时间调整给出了相应设计建议。
This paper introduces the off-axis reflcction system of camera of optical remote sensing satel-lite, the off-axis reflcction system can expand the optical system field of view and improve MTF. But the off-axis angle brings the Problem that integral time is not synchronized. This paper analyzes the factors on asynchrony of integral time of off-axis reflection system for camera. First, a mathematical model of the off-axis angle is set up. Use STK software to simulate the integral time caused by the off-axis. And then use Matlab programming to deal with sampling points. The conclusion is that the larger the off-axis angle is, the smaller the transfer function of the optical system is, and the poorer the quality of the optical imaging. For engineering ap-plication, the paper offers design proposals for adjusting the integral time at the same time.

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本文比较了第一类曲面积分在直角坐标与柱面坐标或球面坐标下的计算方法，给出利用柱面坐标或球面坐标计算第一类曲面积分的公式，并说明在教学过程中，如何让学生易于理解柱面坐标或球面坐标下的面积元素。
This paper compares the first class surface integral calculation method in cylindrical coordinates or Cartesian co-ordinates and spherical coordinates, given the use of cylindrical coordinates or spherical coordinates to calculate surface in-tegral equations of the first kind, and described in the teaching process, how to make students easily understood area element cylindrical coordinates or spherical coordinates.

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目的：研究和探讨老年合并有2型糖尿病的无症状性脑梗死（SBI）患者的头颈部CT血管成像(CTA)的影像学特点及结果分析。方法：65例老年糖尿病合并SBI为观察组，65例非糖尿病合并SBI为对照组，两组均行常规头颈部CTA检查。比较分析两组患者头颈部CTA的特点以及头颈部颅外大血管狭窄程度积分、颅内动脉狭窄积分以及斑块积分。结果：观察组中患者CTA头颈动脉狭窄程度积分、颅内动脉狭窄积分以及斑块积分比对照组均明显增高（P<0.05）。结论：老年糖尿病合并SBI的患者的动脉血管损伤程度高于非糖尿病组，合并有糖尿病的患者更容易出现弥漫性血管粥样硬化性病变，更容易出现多发的颅内血管狭窄。
Objective:To research on iconography characteristic and analysis about the head and neck CT angiography (CTA) in elderly patients with senile brain infarction (SBI) and type 2 diabetes.Methods: 65 cases of elderly diabetic patients with SBI as the observation group, 65 cases of non diabetic patients with SBI as the control group, the two groups were examined by CTA of head and neck . Comparative analysis on characteristic of two groups of patients with CTA of head and neck, the characteristics about the score on large extracranial vascular stenosis, Intracranial artery stenosis and plaque.Results: The score on large extracranial vascular stenosis, Intracranial artery stenosis and plaque in observed people is increased signiifcantly than the control group. (P<0.05).Conclusion:The degree of artery injury in the elderly diabetic patients with SBI is higher than that in the non-diabetic group, the patients with diabetes were more prone to diffuse vascular atherosclerotic lesions,

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随着社会的发展，经济学和数学这两个学科的联系也越来越紧密，利用数学知识去解决经济方面的问题也变得越来越普遍。从现在的经济理论来看，现在用数学知识来分析经济现象分为定量分析和定性分析。然而微积分就是数学的主要内容，而定积分又是微积分的基础，所以利用定积分来解决经济学中的问题也就变得非常的丰富。关于这方面的研究大概也就是一些函数的关系，比如说价格函数、需求函数、收益及成本函数等等，这些都是定积分运用到经济学中的重要的桥梁。要想优化经济的管理和经济的理论，在经济学中，定积分的运用是非常重要的。
With the development of society, the relationship between economics and mathematics becomes closer and closer, and mathematical knowledge is widely used in solving economic problems. Guided by current economy theory, nowadays analysis of economic phenomena with mathematical knowledge is divided into quantitative analysis and qualitative analysis. Calculus is the main part of mathematic while definite integral is the basis of calculus, therefore, it is of great interest to use definite integral to study and solve the economic problems through the links of functions, such as price function, demand function, profit and cost function. It is thus clear that the study of calculus is necessary in economics if we want to optimize the economy management and economy theory.

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