# What does it mean for a region to be simultaneously a region of type 1 and type 2?

I am going through a proof of Green's Theorem for a simple region and I understand the mathematics taking place but do not understand the origins.

'Regions that are simultaneously of type I and II are “nice” regions' is a statement I don't fully understand.

I have proved mathematically that: $$-\iint_G \Bigg(\frac{\partial Q}{\partial x}\Bigg) \cdot dA = \int_{\partial G}Q \cdot dy$$ and $$-\iint_G \Bigg(\frac{\partial P}{\partial y}\Bigg) \cdot dA = \int_{\partial G}P \cdot dx$$ But I do not understand the two types of region being discussed and how a region can be simultaneously both.

Any help understanding this concept would be greatly appreciated.

• You should include the definitions of type I and type II. – eyeballfrog Mar 21 at 21:00

As an example that is Type I but not Type II, consider the region bounded by $$x=0$$, $$x=100\pi$$, $$y=-1$$, and $$y=\sin x$$. It can be decomposed into vertical strips, but horizontal slices would be disconnected.