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 \([\mathcal{R}_1] - [\mathcal{R}_2] - [\mathcal{R}_3] + [\mathcal{R}_4]\).
I know there was already an answer to this question, but it was wrong. I've been trying to do this problem for ages, but I can't get anywhere. Any help is appreciated!
