In many cases, Stokes theorem can result in an integral evaluated by using Green

Question

In many cases, Stokes theorem can result in an integral evaluated by using Green's Theorem. For example, I solved problems #3,5,7,9 at the following page:

http://prntscr.com/2c9kbw

By using Stokes Theorem, then applying Green's Theorem, the problem's were greatly simplified (ps please do not provide a detailed solution as I've already solved these problems!). No need to calculate the normal vector, cross-product, etc. However in some problems, Green's Theorem produces the incorrect answer, why? (I only encountered one case and it can be seen as #12 on the previous link). The choice is obvious, if Green's theorem can be applied, the compution becomes trivial. The definition(s) are more tedious but always works. What's the obvious choice? In terms of #12 seen above, why does Green's Theorem product an incorrect answer?

Thnx a million.

Explanation / Answer

the green's theorem is only applicable when the curve is simple closed curve and hence should be only used at that time

while integration can always assures a correct answer, green's theorem is only applicable for simple closed curve

so my advice first check if the curve is simple closed or not then apply green's theorem

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