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# Let denote the circular region bounded by x^2 + y^2 = 36. The lines x = 4 and y = 3 partition into four regions . Let

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Let $$\mathcal{R}$$ denote the circular region bounded by x^2 + y^2 = 36. The lines x = 4 and y = 3 partition $$\mathcal{R}$$ into four regions $$\mathcal{R}_1, \mathcal{R}_2, \mathcal{R}_3, and \mathcal{R}_4$$. Let $$[\mathcal{R}_i]$$ denote the area of region $$\mathcal{R}_i$$. If $$[\mathcal{R}_1] > [\mathcal{R}_2] > [\mathcal{R}_3] > [\mathcal{R}_4]$$, then compute $$[\mathcal{R}_1] - [\mathcal{R}_2] - [\mathcal{R}_3] + [\mathcal{R}_4]$$.

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Let $$\mathcal{R}$$ denote the circular region bounded by x^2 + y^2 = 36.
The lines x = 4 and y = 3 partition $$\mathcal{R}$$ into four regions $$\mathcal{R}_1, \mathcal{R}_2, \mathcal{R}_3, \text{ and } \mathcal{R}_4$$.
Let $$[\mathcal{R}_i]$$ denote the area of region$$\mathcal{R}_i$$.
If $$[\mathcal{R}_1] > [\mathcal{R}_2] > [\mathcal{R}_3] > [\mathcal{R}_4]$$, then compute $$[\mathcal{R}_1] - [\mathcal{R}_2] - [\mathcal{R}_3] + [\mathcal{R}_4]$$.

$$\text{Let  \mathcal{R}_1 = \mathcal{R}_{11} + \mathcal{R}_{12} + \mathcal{R}_{13} + \mathcal{R}_{14}  } \\ \text{Let  \mathcal{R}_2 = \mathcal{R}_{21} + \mathcal{R}_{22}  } \\ \text{Let (1)\qquad \mathcal{R}_2 = \mathcal{R}_{13} + \mathcal{R}_{14}  } \\ \text{Let (2)\qquad \mathcal{R}_3 = \mathcal{R}_{12} + \mathcal{R}_{13}  } \\ \text{Let (3)\qquad \mathcal{R}_4 = \mathcal{R}_{21}  } \\ \\ \text{Let (4)\qquad \mathcal{R}_{13} = \mathcal{R}_{21}  } \\ \text{Let (5)\qquad \mathcal{R}_{1} -\mathcal{R}_{13}-\mathcal{R}_{14} = \mathcal{R}_{12} + \mathcal{R}_{11}  }$$

$$\begin{array}{|rcll|} \hline [\mathcal{R}_1] - [\mathcal{R}_2] - [\mathcal{R}_3] + [\mathcal{R}_4] &=& ([\mathcal{R}_1] - \underbrace{[\mathcal{R}_2])}_{= [\mathcal{R}_{13}] + [\mathcal{R}_{14}]} - (\underbrace{[\mathcal{R}_3]}_{=[\mathcal{R}_{12}] + [\mathcal{R}_{13}] } - \underbrace{[\mathcal{R}_4]}_{=[\mathcal{R}_{21}] } ) \\\\ &=& ( \underbrace{[\mathcal{R}_1] - [\mathcal{R}_{13}] - [\mathcal{R}_{14}]}_{=[\mathcal{R}_{12}] + [\mathcal{R}_{11}] } ) -( [\mathcal{R}_{12}] + \underbrace{[\mathcal{R}_{13}]}_{=[\mathcal{R}_{21}]} - [\mathcal{R}_{21}] ) \\ \\ &=& ( [\mathcal{R}_{12}] + [\mathcal{R}_{11}] ) -( [\mathcal{R}_{12}] + [\mathcal{R}_{21}] - [\mathcal{R}_{21}] ) \\ \\ &=& ( [\mathcal{R}_{12}] + [\mathcal{R}_{11}] ) -[\mathcal{R}_{12}] \\ \\ &=& [\mathcal{R}_{11}] \\ \\ &=& 8 \times 6 \\ \\ &\mathbf{=}& \mathbf{ 48 } \\ \hline \end{array}$$

This last region is simply a rectangle of height 6 and width 8, so its area is 48.

Jul 6, 2018
edited by heureka  Jul 6, 2018