Area bounded by Floor function
Find the area in sq units of the set of points which satisfy:
for
where denotes greatest integer function
The answer is 8.
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The only values that we can get for [ x ] 2 using the domain are 1, 4, 9, and 16. Since [ y ] 2 can only be equal to 1, 4, 9, and 16. Therefore the only possible values for [ y ] are -4, -3, -2, -1, 1, 2, 3, 4.
This means that
For x = [ 1 , 2 ) , y = [ − 1 , 0 ) or [ 1 , 2 ) .
For x = [ 2 , 3 ) , y = [ − 2 , − 1 ) or [ 2 , 3 ) .
For x = [ 3 , 4 ) , y = [ − 3 , − 2 ) or [ 3 , 4 ) .
For x = [ 4 , 5 ) , y = [ − 4 , − 3 ) or [ 4 , 5 ) .
The graphs of these solutions were represented by 8 squares of side 1. Therefore the area of the set of points which satisfy the equation is 8( 1 x 1) = 8