Guys, can you help me to solve this problem? I don't know how to calculate the area of darkened. If you don't mind, just give me a step to solve this, I am grateful. FYI, the diameter of small circle is 2, the diameter of two congruent circle is 4, and the biggest one is 6.
It's trivial to find the areas of the four circles, so your problem is just subtracting the areas of the six circular segments. Here's two:
$AE=AB=AD=6$, and $BE=BD=4$. Using the Law of Cosines, we have $$BE^2=AE^2+AB^2-2\cdot AE\cdot AB\cdot\cos\angle EAB$$ from which we get $m\angle EAB\approx 0.6797$ or $m\angle EAD\approx 1.3594$. (All angle measures are in radians.) The area of the circular sector $AED$ is $\frac12\cdot AE^2\cdot m\angle EAD=24.4683$, and the area of $\triangle AED=\frac12\cdot AE^2\cdot\sin\angle EAD=17.5993$, which leads to $6.869$ as the area of the circular segment defined by circular arc $EBD$.
Repeat that for the rest of the segments, and you're all set.