How to compute the area of the shadow? 
If we can not use the integral, then how to compute the area of the shadow?
It seems easy, but actually not?
Thanks!

EDIT: Sorry this has an error in it... I solved with a $2E$ rather than a $3E$.
Denote the corners of this unit square by $X,Y,Z,W$ and draw the circles of unit radii with centres at $X$, $Y$, $Z$ and $W$.
If you do this your square is divided up into 21 regions. Denote the central region by $C$. Denote the arrow heads by $E$. Denote the 'other bits' at the edge by $D$, the things that look like annulli by $A$ and the other bits --- the round hats by $B$:

Now the total area is
$$4A+4B+C+8D+4E\overset{!}{=}1,$$
the area of the little circle is
$$4A+4B+C\overset{!}{=}\pi\left(\frac{1}{2}\right)^2,$$
and the area of a quarter of one of those unit circles is
$$3A+2B+C+4D+3E\overset{!}{=}\frac{1}{4}\pi(1)^2.$$
Now solve these equations for $A$ and $B$ in terms of $C$ which should be possible. Now the area you are looking for is $A+2B$... it turns out this EDIT DOES HAVE $E$ dependence and you are left with
$$\frac{4-\pi}{8}+E.$$
