[教育學術] 數學問題

Assume the centre of the sq is at the origin (0,0)
so the vertices of the sq are A (1,1), B (-1,1), C (-1, -1) & D (1,-1)
join the vertices so that the diagonals divide the sq into 4 triangles
consider only the upper triangle formed by (0,0), A and B
coz by symmetry it's easy to find the area of 1 and multiply it by 4 to get the answer

in the upper triangle, all the points inside are closer to the "upper side" AB
than to the other sides

consider a point (x,y), to satisfy the given criteria, we hv

root (x^2 + y^2) < (1-y)
x^2 + y^2 < 1 -2y + y^2
2y < - x^2 + 1
y < (- x^2 + 1) / 2

consider the graph y = (- x^2 + 1) / 2
the desired region is that UNDER this graph and in the specified triangle
that is the area between the lines OA (y = x) and OB (y = - x) and the graph
the intersection points of lines and area of region is to be determined
by solving simultaneous equations and definite integral afterwards, which
i suppose u can do it by yourself, ok? ^_^