Answer
The area of the region
$$
S=\left\{(x, y) | x \geq 0, y \leq 1, x^{2}+y^{2} \leq 4 y\right\}
$$
is equal to $$ \frac{2 \pi}{3}-\frac{\sqrt{3}}{2}$$
Work Step by Step
The area of $S$ can be determined without Calculus as follows:
Note that $$ \theta=\angle C A B=\frac{\pi}{3},$$
so the area is
$$
\begin{split}
S & =\text { area of sector } O A B)-(\text { area of } \triangle A B C)
\\
& = \frac{1}{2}\left(2^{2}\right) \frac{\pi}{3}-\frac{1}{2}(1) \sqrt{3} \\
&=\frac{2 \pi}{3}-\frac{\sqrt{3}}{2}
\end{split}
$$
So the area of the region
$$
S=\left\{(x, y) | x \geq 0, y \leq 1, x^{2}+y^{2} \leq 4 y\right\}
$$
is equal to $$ \frac{2 \pi}{3}-\frac{\sqrt{3}}{2}$$
