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Let \(\mathcal{R}\) be the circle centered at \((0,0)\) with radius 10. The lines \(x = 6\) and \(y=5\) divide \(\mathcal{R}\) 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 find \([\mathcal{R}_1] - [\mathcal{R}_2] - [\mathcal{R}_3] + [\mathcal{R}_4]\).

 Mar 29, 2020
 #1
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By calculus,

\[[\mathcal{R}_1] = 30 + \frac{1}{4} \cdot \pi \cdot 10^2 + \int_0^5 \sqrt{100 - x^2} \ dx + \int_0^6 \sqrt{100 - x^2} \ dx.\]

We can write out the areas similarly, to get [R_1] - [R_2] - [R_3] + [R_4] = 80.

 Mar 29, 2020
 #2
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Just a picture for those who want to see:

AnExtremelyLongName  Mar 29, 2020
 #3
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i solved it already, but thanks for trying to help! the answer was 120.

 Mar 30, 2020

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