<s>[INST] How to find the shaded area How to find the shaded area crossed by semi-circle of radius 2 and quarter-circle of radius 4?

 [/INST] Hint.

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Solution.

$$\begin{align}
\frac12\cdot\text{target area} &= A + B \\
&= \left( \frac12 \cdot (2s)^2\cdot\alpha - 4 C\right) + \left(\frac12 \cdot s^2 \cdot \beta - C \right) \\[4pt]
&= \frac12 s^2\left( 4 \alpha + \beta \right) - 5 C \\[4pt]
&=\frac12 s^2 \left( \frac\pi2 + 3\alpha \right)-5 \cdot \frac{1}{20}(2s)^2 \\[4pt]
&= \frac12 s^2 \left( \frac\pi2 + 3\operatorname{atan}\frac{1}{2} - 2 \right)
\end{align}$$
So, the target area, with $s = 2$, is

$$2 \pi + 12 \operatorname{atan}\frac{1}{2} - 8 $$

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