punts enters de una regió

x^{2}+y^{2} [< r^{2}


y [< ( r^{2}+(-1)x^{2} )^{(1/2)}
x [< ( r^{2}+(-1)y^{2} )^{(1/2)}


x+y [< ( r^{2}+(-1)y^{2} )^{(1/2)}+( r^{2}+(-1)x^{2} )^{(1/2)}


si x=y ==>
2y^{2} [< r^{2} <==> y [< (r/2^{(1/2)})
2x^{2} [< r^{2} <==> x [< (r/2^{(1/2)})


si ( x=0 or y=0 ) ==>
y [< r
x [< r


sum( [x+y] ) [<
...1+...
...4·r+...
...4·sum[y=0-->(r/2^{(1/2)})]( [( r^{2}+(-1)y^{2} )^{(1/2)}] )+...
...4·sum[x=0-->(r/2^{(1/2)})]( [( r^{2}+(-1)x^{2} )^{(1/2)}] )+...
...(-4)[(r/2^{(1/2)})]


x > 0 & y > 0 & xy [< n


y [< (n/x)
x [< (n/y)


x+y [< (n/y)+(n/x)

si x=y ==>
y^{2} [< n <==> y [< n^{(1/2)}
x^{2} [< n <==> x [< n^{(1/2)}


sum( [x+y] ) [< ...
...sum[(y > 0)-->n^{(1/2)}]( [(n/y)] )+...
...sum[(x > 0)-->n^{(1/2)}]( [(n/x)] )+...
...(-1)[n^{(1/2)}]