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!

Let $a$ be a side of the square. Consider the following diagram

The area we need to calculate is as follows.
$$\begin{eqnarray} \color{Black}{\text{Black}}=(\color{blue}{\text{Blue}}+\color{black}{\text{Black}})-\color{blue}{\text{Blue}}. \end{eqnarray}$$
Note that the blue area can be calculated as
$$\begin{eqnarray}\color{blue}{\text{Blue}}=\frac14a^2\pi-2\cdot\left(\color{orange}{\text{Yellow}}+\color{red}{\text{Red}}\right).\end{eqnarray}$$
We already know most of the lengths. What's stopping us from calculating the black area is lack of known angles. Because of symmetry, almost any angle would do the trick.
It's fairly easy to calculate angles of triangle $\begin{eqnarray}\color{orange}{\triangle POA}\end{eqnarray}$, if we use cosine rule.
$$\begin{eqnarray}
|PA|^2&=&|AO|^2+|PO|^2-2\cdot|AO|\cdot|PO|\cos\angle POA\\
a^2&=&\frac{a^2}{4}+\frac{2a^2}{4}-2\cdot\frac a2\cdot\frac{a\sqrt2}{2}\cdot\cos\angle POA\\
4a^2&=&3a^2-2a^2\sqrt2\cos\angle POA\\
1&=&-2\sqrt2\cos\angle POA\\
\cos\angle POA&=&-\frac{1}{2\sqrt2}=-\frac{\sqrt2}{4}.
\end{eqnarray}$$
Now, because of symmetry, we have $\angle POA=\angle POB$, so $\angle AOB=360^\circ-2\angle POA$. So the cosine of angle $\angle AOB$ can be calculated as follows:
$$\begin{eqnarray}
\cos\angle AOB&=&\cos(360^\circ-2\angle POA)=\cos(2\pi-2\angle POA)\\
\cos\angle AOB&=&\cos(-2\angle POA)=\cos(2\angle POA)\\
\cos\angle AOB&=&\cos^2(\angle POA)-\sin^2(\angle POA)\\
\cos\angle AOB&=&\cos^2(\angle POA)-(1-\cos^2(\angle POA))\\
\cos\angle AOB&=&2\cos^2(\angle POA)-1\\
\cos\angle AOB&=&2\cdot\left(-\frac{\sqrt2}{4}\right)^2-1=-\frac34\\
\end{eqnarray}$$
From this, we can easily calculate the sine of angle $\angle AOB$, using Pythagorean identity.
$$ \sin\angle AOB=\sqrt{1-\frac9{16}}=\sqrt\frac{16-9}{16}=\frac{\sqrt7}4 $$
Going this way, I believe it's not hard to calculate other angles and use known trigonometry-like formulas for area. Then you can easily pack it together using the first equation with colors.
