而积分函数就是dxdy前的这个函数，它与积分边界无关。
While the function that you are integrating goes before the dx dy and not into the bounds or anything like that.
本文定义了象差积分函数，导出了形式简洁的全部二级象差函数。
By defining integration function, all second-order aberration coefficients have been expressed in compact form.
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.
该定积分可用MATLAB提供的积分函数计算，或者利用蒙特卡罗方法通过编程求解。
The definite integral can be calculated with the integration function in MATLAB or can be solved by using the Monte Carlo programming method.
该法与分段卷积法和图解卷积法相比，不仅可直接表达积分函数，而且又能简化卷积计算。
Compared with the sectioned and graphic convolutions, the method is able not only to express directly the integral functions but to simplify the calculation.
本文对电磁场问题的理论计算中常遇到的含有指数函数和指数积分函数的积分，得出了解析表达式。
The analytical expressions are derived for the integrals containing exponential and exponential integral which appear frequently in electromagnetic field problems.
利用上方法计算了汉中地区农科所的水稻产量及光温资料，所得光温回归积分函数曲线与汉中盆地水稻生产实际相吻合。
Regressive integration curve based on thermal and solar light data and rice yield from Han-Zhong Institute Agricultural Science is coincident with the actual rice production of Hanzhong basin.
要懂得如何计算一个函数的二重积分。
二重积分可以用来得到关于这个区域的某些信息，或者这个区域上的一个函数的平均值，等等。
We can use that to get information about maybe the region or about the average value of a function in that region and so on.
那么我就能名正言顺地,用R上的某个函数的二重积分来替代通量的线积分。
Then I can actually -- --replace the line integral for flux by a double integral over R of some function.
我要说的是取其一，积分得到包括另一变量的函数的结果，然后对结果求导，进行比较看得到什么。
But what I am saying is just take one of them, integrate, get an answer that involves function of the other variable, then differentiate that answer and compare and see what you get.
对函数1进行积分。
就二重积分来讲,它是对区域里函数值求总和。
The way we actually think of the double integral is really as summing the values of a function all around this region.
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.
一般cp和，都是温度的函数，因此实际上，我们可以将这个积分计算出来。
Cv So, for Cp and Cv, these are often quantities that are measured as a function of temperature, and one could, in fact, calculate this integral.
计算质量可看成二重或三重积分，这取决于空间维数，取决于密度函数，dA还是。
And mass will be double or triple integral, depending on how many dimensions you have, dV of whatever density function you have, dA or dV.
区域R的面积是函数1在R上的二重积分。
那么，你会说那很简单，积分就可以了，除非你不知道函数是什么。
So, you'd say, oh, it's easy. Let's just integrate, except you don't have a function.
就是x乘以函数做积分的均值，区域中的。
可能是积分边界容易确定，但函数可能就比较困难，因为它含有这些sin,cos在里面。
Maybe it will be easier to set up bounds but maybe the function will become harder because it will have all these sines and cosines in it.
无论函数是什么，如果你的函数除了原点处处为，在原点是其他的值，那积分还是0的。
no matter what value you put for a function, 0 if you have a function that's zero everywhere except at the origin, and some other value at the origin, the integral is still zero.
你们都知道，要求这个面积是非常简单的，如果你们想的话，可以积分除了椭圆函数以外的任何函数。
So, you know, if you find that the area is too easy, you can integrate any function other than ellipse, if you prefer.
我们有这个积分常数，但是除去它，我们应该从这里能得到一个势函数。
We have this integration constant, but apart from that we know that we should be able to get a potential from this.
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.
微积分基本定理，不是曲线积分的，告诉我们，如果对函数的导数积分，就会得回原函数。
So, the fundamental theorem of calculus, not for line integrals, tells you if you integrate a derivative, then you get back the function.
但是微积分，主要是学习函数的。
一旦需要计算这个积分，只需要计算这个函数的三重积分。
Once you have computed what this guy is, it's really just a triple integral of the function.
它说，如果你对一个函数的梯度做线积分，就能得到原函数。
It tells you, if you take the line integral of the gradient of a function, what you get back is the function.
作为引子，你们可能已经知道了，一元微积分里面的一个小把戏，也就是求隐函数微分法。
And, just to motivate that, let me remind you about one trick that you probably know from single variable calculus, namely implicit differentiation.
如果有一个梯度场，然后想计算线积分，我知道积分值是,起点的函数值减去终点的函数值。
Well, if I have a gradient field, then if I try to compute this line integral, I know it will be the value of the function at the end point minus the value at the starting point.
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