MHB What is the limit of a bounded region with a specific boundary?

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The discussion focuses on determining the area \(A_r\) of a bounded region defined by the boundary equation \((6y^2r-x)(6\pi^2 y-x)=0\). Participants are tasked with calculating the limit $$ \lim_{n \to \infty}(\sqrt{A_1 A_2 A_3}+\sqrt{A_2 A_3 A_4}+\cdots \text{n terms}).$$ To solve this, one must first derive the expression for \(A_r\) based on the given boundary conditions. The conversation emphasizes the importance of understanding the geometric implications of the boundary equation in relation to the area calculations. Ultimately, the limit's evaluation hinges on the accurate computation of the areas \(A_r\).
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Let $A_r(r \in \mathbb{N})$ be the area of the bounded region whose boundary is defined by $(6y^2r-x)(6\pi^2 y-x)=0$ then find the value of

$$ \lim_{n \to \infty}(\sqrt{A_1 A_2 A_3}+\sqrt{A_2 A_3 A_4}+\cdots \text{n terms})$$
 
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Start by finding \(A_r\).
 
Thread 'Problem with calculating projections of curl using rotation of contour'

Thread 'Problem with calculating projections of curl using rotation of contour'

Hello! I tried to calculate projections of curl using rotation of coordinate system but I encountered with following problem. Given: ##rot_xA=\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}=0## ##rot_yA=\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}=1## ##rot_zA=\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}=0## I rotated ##yz##-plane of this coordinate system by an angle ##45## degrees about ##x##-axis and used rotation matrix to...
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