为更好地指导教学，文章还对如何准确、有效地利用对称性简化二重积分的计算作了进一步的探讨。
To give better advice on teaching, futher discussion, is made on how to simplify the double integral operation with symmetry, precisely and effectively.
至此我已经得到了，用来计算二重积分的所有量。
要懂得如何计算一个函数的二重积分。
我们已经做过的一个例子是，计算四分之一单位圆上的二重积分。
One example that we did, in particular, was to compute the double integral of a quarter of a unit disk.
不管是线积分或是二重积分，也不管它们表示的是功还是通量，计算它们的方法实际上是一样的。
And whether these line integrals or double integrals are representing work, flux, integral of a curve, whatever, the way that we actually compute them is the same.
要计算二重积分，要做的就是要利用切面。
So, to compute this integral, what we do is actually we take slices.
当计算二重积分时，要多了解这些符号的具体含义。
OK, we'll come up with more concrete notations when we see how to actually compute these things.
用格林公式计算…，只是计算…，让我们忘记…，应该是，算沿闭曲线的线积分值，可以通过二重积分来算。
To compute things, Green's theorem, let's just compute, well, let us forget, sorry, find the value of a line integral along the closed curve by reducing it to double integral.
同时还指出了几本高等数学参考书中关于不定积分、二重积分、曲线积分计算中出现的错误。
And it also points out the mistakes in some Higher Mathematics Reference Books concerning indefinite integral, double integral and curve integral calculations.
从理论和实例两个方面，指出了对称区域上二重积分计算中易出现问题的原因，并给出了避免错误的方法。
From the theory and examples, the article points out why in the symmetric block the calculation of the double integral is easy to go wrong and offers some methods to avoid errors.
从理论和实例两个方面，指出了对称区域上二重积分计算中易出现问题的原因，并给出了避免错误的方法。
From the theory and examples, the article points out why in the symmetric block the calculation of the double integral is easy to go wrong and offers some methods to avoid errors.
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