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来想想二重积分的其他用途吧。
Let's think about various USES of double integrals.
youdao
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先来说说关于建立二重积分的事。
So, the first topic will be setting up double integrals.
youdao
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这是建立三重积分的绝佳练习。
See, this is actually good practice to remember how we set up triple integrals.
youdao
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这也就是二重积分的意义。
That is what a double integral means.
youdao
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这和计算其他三重积分的方法是相同的。
It's just the same way that you would compute any other triple integral.
youdao
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在化成两个积分的情况下，如何处理呢？
So, what I do in that case is I just make two integrals.
youdao
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这是由于我们才刚开始做二重积分的缘故。
That is because we are only doing double integrals so far.
youdao
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你们应该懂很多积分的。
You know that much about integrals.
youdao
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怎么变换积分的顺序呢？
So, how do we exchange the order of integration?
youdao
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请记住，这里主要的技巧是找到积分的界限。
So, remember, the main trick here is to find the bounds of integration.
youdao
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线积分的意义，就是这力所做的功。
The line integral was actually the work done by the force.
youdao
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在这时这个问题中微积分的部分。
So that's the calculus part.
youdao
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这就是二重积分的基本定义了。
That's the basic definition.
youdao
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我们已经学过，如何建立这种二重积分的公式。
We've seen various formulas for how to set up the double integral.
youdao
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至此我已经得到了，用来计算二重积分的所有量。
Now I have everything I need to compute my double integral.
youdao
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那么，接下来我们来求积分的上下限吧。
So, let's try to find the bounds.
youdao
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为微积分的创立做出了贡献。
For the creation of calculus to make a contribution.
youdao
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我们可利用积分的可加性。
We can use the additive property of the integral.
youdao
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今天这个是用来求F的法向分量的积分的。
The one today is for integrating the normal component of F.
youdao
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当然，可以把不影响积分的常数a提出来。
And, of course, I can get rid of some a's in there and take them out.
youdao
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就是x乘以函数做积分的均值，区域中的。
x It's the average value of a function, x in the region.
youdao
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包括积分的获取和消费。
Include to be gotten integrally and be consumed.
youdao
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讨论了微积分的产生对博弈论产生的影响。
The effect of the appearance of calculus on game theory.
youdao
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本文就非正常积分的定义提出了一些看法。
Viewpoint which are about definition of the non-normal integration are given in this paper.
youdao
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你们应该有一本关于多变量微积分的好书。
You should have a good book on multivariable calculus.
youdao
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新数据显示，这些人构成了积分的11.9%。
The new data shows that this group comprised 11.9 percent of scores.
youdao
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利用欧拉积分的特殊性质，使求解可行且有效。
By using the special quality of Euler integrals, we can solve the question much better.
youdao
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我积分的是f，y
What I'm integrating to is f of x of t and y of t.
youdao
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我积分的是f，y
What I'm integrating to is f of x of t and y of t.
youdao