Pixelated Poké Ball

Algebra Level 5

\[{ x }^{ 2 }+\left\lfloor { y }^{ 2 } \right\rfloor =20\\ \left\lfloor { x }^{ 2 } \right\rfloor +{ y }^{ 2 }=20\]

Find the area bounded by the two Cartesian equations above. If this area can be expressed in the form \[\sum _{ n=1 }^{ Y }{ R\sqrt { n } } \left( \sqrt { G-n } -\sqrt { B-n } \right)\] with \(R\), \(G\), \(B\), and \(Y\) composite numbers, write your answer as \(\dfrac { R\times G\times B }{ Y }\).

Bonus: Using this sum, generalize the bounded area with respect to \(r\): \[{ x }^{ 2 }+\left\lfloor { y }^{ 2 } \right\rfloor ={ r }^{ 2 }\\ \left\lfloor { x }^{ 2 } \right\rfloor +{ y }^{ 2 }={ r }^{ 2 }\]


This problem was created in celebration of Pokémon's 20th anniversary.
The picture of the graph was produced from Desmos.
Image Credit: Pokeball by phantompanther, DeviantArt.
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