Is there a way to calculate the area of this intersection of four disks without using an integral? Is there anyway to calculate this area without using integral ?


Let $R$ be its radius and $D$ its diameter: $R = 5$, $D = 10$.
$$\begin{align}
\text{Area of big square} &= D^2 = 100 \\
\text{Area of circle} &= \frac{\pi D^2}{4} \approx 78.54 \\
\text{Area outside circle} &= 100 - 78.54 = 21.46 \\
\text{Area of 4 petals} &= 78.54 - 21.46 = 57.08 \\
\text{Area of single petal} &= \frac{57.08}{4} = 14.27 \\
\text{Area of small square} &= R^2 = 25
\end{align}$$
Let $x$ denote the area of the portion selected.
$$\begin{align}
\text{Area of 2 petals} &= 2 \cdot 14.27 = 28.54 \\
0 &= 25 - 28.54 + x  \\
\text{Area of 8 petals} &= 2 \cdot 57.08 = 114.16 \\
\text{OR}\\
100 - 114.16 + 4 x &= 0
\end{align}$$
$$ x = 3.54 $$
Here we have a small area which needs to be added,
that I found out by modeling to be $4.34$
$$\begin{align}
\text{This gives us the desired area of}\\ 
\text{4.34+3.54} &\approx 7.88\\
\text{& a percentage of} &\approx 4 \times 7.88\\
\text{that is} &\approx 31.52\%
\end{align}$$
