给出了在各种情况下积分第二中值定理“中间点”的渐近性的几个结论，相信在积分学中有着很重要的作用。
In this paper, second mean value theorem for integrals is studied, and some results of the inverse problem of the theorem are obtained.
但在大多数情况下，我们需要了解怎样建立二重积分。
But most of the time we need to learn how to set up double integrals.
如果你还不是完全清楚，建议你去复习一下，如何建立二重积分。
OK, if that's not completely clear to you, then I encourage you to go over how we set up double integrals again.
一种考虑这个问题的办法是，如果你还觉得，二重积分是求体积的话，那这个度量的，就是函数1的图形下的体积。
One way to think about it, if you're really still attached to the idea of double integral as a volume what this measures is the volume below the graph of a function one.
不过绝大多数情况下，都需要知道怎么去计算，建立与计算积分。
But, mostly you should know how to compute, set up and compute these things.
通常，你只会在这两种情况下用替代法，或者这可以极大地简化被积函数，或者就是这可以简化，积分的上下限。
I mean, normally, you would only do this kind of substitution if either it simplifies a lot the function you are integrating, or it simplifies a lot the region on which you are integrating.
再一次申明一下，不要关掉你付款的信用卡，这样会降低你的信誉积分！
Again, do NOT close paid off credit CARDS because it will lower your credit score!
这是极坐标系下的二重积分。
我指的是，在这种形式下，它和一元微积分的表述是一样的。
I mean, in this form, actually it's the same statement as in single variable calculus.
在化成两个积分的情况下，如何处理呢？
请说，你想知道极坐标系下的积分边界，这是一个二重积分。
Yes? In case you want the bounds for this region in polar coordinates, indeed it would be double integral.
设想一下，有一天你能够用飞行里程积分换杯啤酒喝。
Imagine someday paying for a beer with frequent flier miles.
通常你转换到极坐标系下，有可能积分区域更容易建立。
I mean usually you will switch to polar coordinates either because the region is easier to set up.
那么我们将不会建立带上下限的积分，和积分dx,dy, dy, dx或者在极坐标做线积分，需要说，让我们回忆一下质心的定义。
So, instead of setting up the integral with bounds and integrating dx dy or dy dx or in polar coordinates, I'm just going to say, well, let's remember the definition of a center of mass.
那么，改变一下区域，把这块移到另一侧，我们得到对xdA的双重积分,是原点那的圆面积乘质心的x的坐标。
So, switching the area, moving the area to the other side, I'll get double integral of xdA is the area of origin times the x coordinate of the center of mass.
相对简单的、找出边界的办法,就是在u、v坐标系下画出积分区域。
And to find the bounds perhaps the easiest is to draw a picture of a region in u, v coordinates.
在两个俱乐部还剩两场比赛的情况下，热刺暂以1分的优势位居积分榜第四。
Tottenham are just a point ahead in fourth place with two games remaining for each club.
nds的线积分，So，，line，integral，of，F。n，ds,是沿着C的线积分,需要确认一下，我写的是正确的。
F becomes the integral along C just to make sure that I give it to you correctly.
我能不能借一下你的微积分课本？我把我的书忘在教室里了。
Can I borrow your calculus textbooks I left mine in the classroom.
在给定的插棒顺序下，用硼补偿法作了微分和积分价值刻度。
The differential and integral worths were calibrated by boron compensation method in given insertion sequence.
在少赛一场的情况下，斑马军团重新回到积分榜榜首，与拉齐奥并驾齐驱。
The Bianconeri are now back at the top of the table alongside Lazio with a game in hand.
变限积分函数的性质主要由被积分函数的性质、积分上(下)限的结构来决定。
The variation range integral function nature mainly by the structure which by in the integral function nature, the integral (next) is limited decided.
同时建立了另一类上半平面解析函数在其分布边值意义下的积分公式。
And the distributed boundary value's integral formula of another set of analytic function in upper half-plane is established.
我能不能借一下你的微积分课本？
证明了在介质为分区均匀的条件下，这个微分-积分方程的解满足不同介质边界面的边界条件。
It is proved that if the medium is piecewisely homogeneous, the solution of the differential -integral equation satisfies the boundary conditions on boundaries between different media.
描述了短曝光情况下，补偿效率积分概率的意义。
Integral probability of compensation efficiency Strehl ratio is given in the sense of short exposure.
本文由力学的普遍原理——虚功原理推导出了三维情况下J积分守恒的一般形式；
The general form of J-integral conservation law in three dimensions has been derived from the principle of virtual work.
结果表明，不同构象下各体系有效交换积分值都小于零，自由基间表现为反铁磁性耦合。
The results show that the effective exchange integral values are less than zero under different conformations, so the radicals display the anti-ferromagnetic coupling.
在一定条件下，对振荡积分值做贡献的主要来自被积函数在临界点附近的小区间的积分值。
Under some conditions, the major contribution to the value of oscillatory integral comes from neighborhoods of those critical points of the integrand.
在一定条件下，对振荡积分值做贡献的主要来自被积函数在临界点附近的小区间的积分值。
Under some conditions, the major contribution to the value of oscillatory integral comes from neighborhoods of those critical points of the integrand.
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