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

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The discussion centers on proving a mathematical statement regarding a simple regular curve C that encloses a region R with area k. The proof utilizes Green's theorem to establish that the integral of a specific linear combination of x and y over the curve equals (b1 - a2) multiplied by the area k. The calculation involves applying the theorem to relate the line integral to a double integral over the region R. The conclusion confirms that the expression holds true, demonstrating the relationship between the constants and the area. This proof effectively illustrates the application of Green's theorem in geometric contexts.
Fernando Revilla
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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.
 
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Using the Green's theorem: $$\int_C(a_1x+a_2y+a_3,b_1x+b_2y+b_3)d\alpha=\iint_R\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy\\=\iint_R(b_1-a_2)dxdy=(b_1-a_2)\iint_Rdxdy=(b_1-a_2)k$$
 

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