The shaded region below is the common area to four semicircles whose diameterare the sides of the square with side length 4x. Find the area of the shaded region in terms of x.

Lets divide the square into four parts and take a shaded region and find its area
So the 1/4 square has a side length of 2x, half of the given square. 
Part of this is shaded and the other part is not shaded. 
Let us find the area of the non shaded part (white). 

The shaded part is a quarter of a disk (circle).
area of non shaded area = (2x)2 - (1/4) Pi (2x)2

The area of the non shaded part is 8 times the non shaded area in the present shape which was calculated above.

Area of shaded part     = total area of square - total non shaded area 
            = (4x)2 - 8*[ (2x)2 - (1/4) Pi (2x)2 ]
            = 16x2(Pi/2 - 1)           

Hence the area of the shaded part is 16x2‚Äč (Pi/2  - 1)

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