You know that much about integrals.
你们应该懂很多积分的。
I mean, you are doing single integrals.
我的意思是，做一元积分时。
That we can turn the sums into integrals.
以至于我们可以把求和变成积分。
So, line integrals we know how to evaluate.
我们知道如何计算线积分。
OK, anyway, let's move on to line integrals.
再转到线积分。
Some of them compute different line integrals.
它们中的一些在做不同的线积分。
What if I add all of the small line integrals?
那如果将所有线积分都加起来呢？
I would just calculate three easy line integrals.
我只需计算三个简单的线积分。
You don't have to set up all these line integrals.
不用建立这些线积分。
Let's think about various USES of double integrals.
来想想二重积分的其他用途吧。
Let's start right away with line integrals in space.
现在开始学习空间线积分了。
So conceptually it is very similar to line integrals.
从概念上来看，这与线积分相似。
One is setting up and evaluating double integrals.
一个是建立并计算二重积分。
So, the first topic will be setting up double integrals.
先来说说关于建立二重积分的事。
So, what I do in that case is I just make two integrals.
在化成两个积分的情况下，如何处理呢？
And, surface integrals, we know also how to evaluate.
对于曲面积分，也已经知道如何计算。
The other one is setting up and evaluating line integrals.
另一个是建立并计算线积分。
That is because we are only doing double integrals so far.
这是由于我们才刚开始做二重积分的缘故。
So, in fact, I have to break this into two different integrals.
实际上，要拆成两个不同的积分。
Unfortunately, it is not quite as simple as with line integrals.
很遗憾，它的过程不像线积分那样简单。
The advantage of this method is you don't have to write any integrals.
这种方法的优点是，不用写出任何积分。
But most of the time we need to learn how to set up double integrals.
但在大多数情况下，我们需要了解怎样建立二重积分。
The claim is we are able, to do double integrals in polar coordinates.
这也就说明了，可以用极坐标做二重积分。
An important method for evaluating integrals is a change in variables.
积分值的一种重要方法是变量代换法。
See, this is actually good practice to remember how we set up triple integrals.
这是建立三重积分的绝佳练习。
And, of course, we know also how to set up these integrals in polar coordinates.
当然，我们也知道，怎样用极坐标来计算这些积分。
Next is one of the easier integrals but always seems to cause problems for people.
接下来是比较容易积分之一，但似乎总是引起人们的问题。
OK, so anyway, again, conceptually, we have, really, three different kinds of integrals.
无论如何，再说一次，从概念上说有三个不同种类的积分。
By using the special quality of Euler integrals, we can solve the question much better.
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.
利用欧拉积分的特殊性质，使求解可行且有效。
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