我们知道现在要积分面积元素，那就对dxdy积分。
So, we know that we want to integrate just the area element, let's say, dx dy over the origin inside the ellipse.
该结构利用积分面积相等的原则确定开关角，通过串级叠加的方法得到标准正弦波输出。
In the structure, the (equal) integration area method was used to determine the control switch Angle, and the sinusoidal voltage wave was obtained by means of the concatenate superposition.
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.
如何用二重积分来表示面积呢？
播种面积分配制、价格支持和作物丢弃令美国农业面貌一新。
Acreage allotments, price supports and crop set-asides reshaped the face of American agriculture.
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.
这里使用dS而不是dA，因为我习惯把dA看成是，坐标平面中的面积元，就像在二重积分中所做的那样，待会大家会了解。
I am using dS rather than dA because I still want to think of dA as maybe the area in one of the coordinate planes like the one we had in double integrals. You will see later where this comes in.
举个例子，区域R的面积是dA的二重积分，便于理解，在这里写成。
So, for example, the area of region is the double integral of just dA, 1dA or if it helps you, one dA if you want.
那么积分1dxdy的值,就是单位圆的面积π
Then the integral of one dxdy will just be the area of this unit disk which is pi.
这把一个向量场的线积分，和另外一个向量场的曲面积分联系起来。
This relates a line integral for one field to a surface integral from another field.
平面中的通量和空间中的通量有很大区别，在平面中，通量仅仅是线积分的另一种形式，而在空间中，它表现为曲面积分。
Flux looks quite different in the plane and in space because, in the plane, it is just another kind of line integral, while in space it is a surface integral.
因此在决定选择单桅帆船还是双桅帆船时，决定因素并非帆面积，而是将这个帆面积分布在两面帆或是三面帆上带来的相应差异。
So it's not the sail area that matters when deciding between a sloop and a ketch, but differences resulting from having that area spread over three rather than two sails.
对于曲面积分，也已经知道如何计算。
当我们在做一重积分时，并不是为了去求平面某区域的面积。
When you do single integrals it is usually not to find the area of some region of a plane.
在我花了这么多时间来告诉你们，如何计算曲面积分之后，我打算告诉你们，如何避免计算它们。
After spending a lot of time telling you how to compute surface integrals, now I am going to try to tell you how to avoid computing them.
城市周围大面积分布的泥炭沼泽已经干涸着火，将莫斯科笼罩在令人窒息的烟雾中。
Vast stretches of peat bog surrounding the city have dried out and caught fire covering Moscow with choking smog.
比如，微积分是从斜率，面积和变化率等等话题中引入的。
For instance, calculus is usually first introduced interms of slopes, areas, rates of change, and so forth.
它是一个面积分，所以这里需要两个变量。
It is a surface integral, so we need to have two variables in there.
计算这个曲面积分的方法，和其他任何曲面积分的一样。
And, the way in which you would compute the surface integral is just as with any surface integral.
每当做通量的曲面积分时，你要做两件事。
Whenever you do a surface integral for flux you have two parts of the story.
她说，这些传感器大面积分散各处，而且需要传输大量数据。
She says that these sensors are spread out over large areas and need to transmit large amounts of data.
也就是最终要摆脱曲面积分，回到常规的二重积分。
And this is finally where I have left the world of surface integrals to go back to a usual double integral.
这个曲面积分，不是同一个向量场。
OK, and that surface integral, well, it's not for the same vector field.
所以要用面积分，而非线积分。
And so it will be a surface integral, not a line integral anymore.
共和国的三个都市化岛屿：贝提欧、拜里基和纳尼凯——面积分别增加了30%(36公顷)，16.3%(5.8公顷)，12.5%(0.8公顷)。
The three major urbanised islands in the republic - Betio, Bairiki and Nanikai - increased by 30 per cent (36 hectares), 16.3 per cent (5.8 hectares) and 12.5 per cent (0.8 hectares), respectively.
相反，我想要把它变成面积分。
And instead I will want to reduce things to a surface integral.
城镇和农村居民人均居住面积分别达到16平方米和23平方米。
Urban and rural residents per capita living area reached 16 square meters and 23 square meters.
区域R的面积是函数1在R上的二重积分。
如果在这个曲面积分中是以dx,dy,dz结尾的，则表明出现了很大的问题。
If you end up with a dx, dy, dz in the surface integral, something is seriously wrong.
过去四十年来，非洲新增灌溉面积仅为400万公顷，而同期中国和印度的新增面积分别为2500万公顷和3200万公顷。
Over the last 40 years, only 4 million hectares of new irrigation have been developed, compared to 25 and 32 million hectares for China and India respectively.
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