Example 11 As shown in Figure 8, sector $A O B$ is one quarter of a unit circle. The center of the semicircle $\odot O_{1}$, $O_{1}$, is on $O A$ and is internally tangent to $\overparen{A B}$ at point $A$. The center of the semicircle $\odot O_{2}$, $O_{2}$, is on $O B$ and is internally tangent to $\overparen{A B}$ at point $B$. The semicircle $\odot O_{1}$ is tangent to the semicircle $\odot O_{2}$. Let the sum of the radii of the two circles be $x$, and the sum of their areas be $y$.
(1) Try to establish the analytical expression of the function $y$ with $x$ as the independent variable;
(2) Find the minimum value of the function $y$.
(2000, Taiyuan City Junior High School Mathematics Competition)
(1) As shown in Figure 8, let the radii of $\odot O_{1}$ and $\odot O_{2}$ be $R$ and $r$ respectively. Then,
$$
\begin{array}{l}
y=\frac{1}{2} \pi\left(R^{2}+r^{2}\right) \\
=\frac{1}{2} \pi\left[(R+r)^{2}-2 R r\right] .
\end{array}
$$

Connecting $O_{1} O_{2}$, the line connecting the centers must pass through the point of tangency. In the right triangle $\triangle \mathrm{O}_{1} \mathrm{OO}_{2}$, by the Pythagorean theorem, we have
$$
(R+r)^{2}=(1-R)^{2}+(1-r)^{2} \text {, }
$$

which simplifies to
$$
R r+R+r=1 \text {. }
$$

Thus, $y=\frac{1}{2} \pi\left\{(R+r)^{2}-2[1-(R+r)]\right\}$
$$
=\frac{1}{2} \pi\left[(R+r)^{2}+2(R+r)-2\right] \text {. }
$$

Given $R+r=x$, then
$$
y=\frac{1}{2} \pi\left(x^{2}+2 x-2\right) \text {. }
$$
(2) Since $R+r \geqslant 2 \sqrt{R r}$, we have
$$
\frac{1}{4}(R+r)^{2} \geqslant R r \text {. }
$$

Also, $R r=1-(R+r)$, so
$$
(R+r)^{2}+4(R+r)-4 \geqslant 0 \text {. }
$$

Clearly, $R+r>0$.
Thus, $R+r \geqslant 2(\sqrt{2}-1)$, which means
$$
x \geqslant 2(\sqrt{2}-1) \text {. }
$$

Therefore, the function is
$$
y=\frac{1}{2} \pi\left(x^{2}+2 x-2\right) \text {. }
$$

When $x=2(\sqrt{2}-1)$, the minimum value is $(3-2 \sqrt{2}) \pi$.
Note: This problem is a geometric optimization problem. By establishing a quadratic function model, applying the Pythagorean theorem and the properties of tangent circles, we derive the relationship between the radii. Using the arithmetic mean-geometric mean inequality, we find the range of $x$, and thus determine the minimum value.