我所要求的只是替代方案，而不是让我们所有人都走上学习微积分的道路。
All that I ask is that alternatives be offered instead of putting all of us on the road to calculus.
至此我已经得到了，用来计算二重积分的所有量。
所以，只需带一张信用卡。可以携带一张积分或返现多的，或者一张不收取国际手续费的（我倾向于这点）。
Instead, stick to one credit card. Either one that gets you the most points or cashback or one that doesn't charge international fees (my preference).
我只需计算三个简单的线积分。
我证出了分别沿着C1和C2的两个线积分是相等的。
OK, so I proved that my two line integrals along C1 and C2 are equal.
就我这一方面而言，我获得三站冠军，六次杆位和相同的最快圈速。相对我去年的积分有提升。
For my part, I won three RACES and took six pole positions and the same number of race fastest laps, improving on my points score from last year.
我对你们的要求越低，物理老师或其他科老师，就会抱怨的越厉害，他们会抱怨“这些家伙居然不知道怎么求积分
I should say, the more I tell you I don't need you to know, the more your physics and other teachers might complain that, oh, these guys don't know how to integrate.
对我喜欢的曲线，计算其上的线积分，在这条线上所做的功。
And let's take my favorite curve and compute the line integral of that field, you know, the work done along the curve.
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.
我要把我的黑板分成三部分，在这我将要写三重积分。
OK, so I'm going to divide my blackboard into three pieces, and here I will write triple integrals.
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.
也就是，如果我需要计算一个线积分，无论我取怎样的路径,只要路径是从点a到点。
OK, so the claim is if I have a line integral to compute, that it doesn't matter which path I take b as long as it goes from point a to point b.
这是我知道的，唯一可以计算这种积分的方法，令x=sin，θ,那么√，就是cos，θ
That's the only way I know to compute an integral like that, OK?
我来计算ydx+xdy的积分。
这里我说的积分常数，它是不含x的一项。
And here, when I say there is an integration constant, it just means the extra term does not depend on x.
我积分的是f，y
当我通过这边的时候，我就可以进入了积分区域。
什么是我的信用积分？
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.
你或许因此对自己说，趁着我的大脑还没退化，赶紧报名学习史瓦西里文和微积分，还有手风琴班吧！
So you may be saying to yourself, I have to sign up right now for Swahili and calculus and accordion lessons before my brain withers away!
我的意思是，做一元积分时。
但是在这段时间里，我更多使用的是离散数学而不是微积分，所以我需要复习才能够找出答案。
But, I use discrete mathematics much more than calculus these days, so I need to go back and look at how to find the answer.
重新分解积分域。。。,我的目标是，用另外一种形式重写这个积分。
The other way around, I have to, so my goal, now, is to rewrite this as an integral.
我的排放交易网站在自愿的碳市场上以你的名义出售该积分，现行价格为10～25美元，所有交易抽取20%的费用。
My Emissions Exchange sells the credit on your behalf on the voluntary carbon market at a going rate of between $10 and $25, taking a 20 percent cut on all sales.
当然我更希望你们记住啦，但是如果脑海中一片空白的话，如果可以写出积分边界和其他的东西，也是可以得到部分分数的。
Well, I mean, I prefer if you know, but if you have a complete blank in your memory, there will still be partial credit were setting up the bounds and everything else.
有些网站给好的答案送积分，但是有些积分系统比较随意，我至今也不知道积分到底有何价值。
Other sites award points for good answers, although some points systems may be arbitrary and I'm still not sure what value most of those points have.
那么，它的总的线积分,就等于沿着C’，我是指外面这个的线积分。
And, so the total line integral for this thing is equal to the integral along C prime, I guess the outer one.
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.
他补充道：“我很高兴球队现在处在积分榜第四的位置，对小伙子们的表现非常满意。”
He added: "I would be happy with fourth place." I am happy with the play of the team.
他补充道：“我很高兴球队现在处在积分榜第四的位置，对小伙子们的表现非常满意。”
He added: "I would be happy with fourth place." I am happy with the play of the team.
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