Four circles of unit radius are drawn with centers $(1,0)$, $(-1,0)$, $(0,1)$, and $(0,-1)$. a circle with radius 2 is drawn with the origin as its center. what is the area of all points that are contained in an odd number of these 5 circles? (express your answer in the form "a pi + b" or "a pi - b", where a and b are integers.)

Check the picture 1.

The regions contained in an odd number of these 5 circles are

the pink colored region, contained only in the circle with radius 2, 

and the yellow regions, each contained in the intersection of 2 small circles and the third large circle.

So what we need to determine is the "area pink + area yellow".

Consider the second picture.

Let A2 be the area of the right isosceles triangle, and A1 be half of the yellow region.

the region A2 + A1 is called a sector, and it is 1/4 of the area of a unit circle.



Thus, the overall yellow region is 

The purple area is:
 
Area of large circle - 4*(Area of 1 small circle) + yellow region.

(because subtracting all 4 small circles means that each of the 4 separate yellow regions have been subtracted twice)



Finally, the total area is :  (units squared) 

     square units