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Circle Problem

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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