MHB Proving a Statement about a Regular Curve: R of Area k

[Fernando Revilla]

I quote a question from Yahoo! Answers

If C is a simple regular curve that encloses a region R of area k. Prove that if ai, bi are constants
∫(c) [a1x + a2y + a3, b1x + b2y + b3]dα = (b1 - a2)k. thanks

I have given a link to the topic there so the OP can see my response.

 

[Fernando Revilla]

Using the Green's theorem: $$\int_C(a_1x+a_2y+a_3,b_1x+b_2y+b_3)d\alpha=\iint_R\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy\\=\iint_R(b_1-a_2)dxdy=(b_1-a_2)\iint_Rdxdy=(b_1-a_2)k$$