Solution
= Solution
Use the dimensionless coordinate $u=x/L$:
$$
P=\int_{0.45}^{0.55}2\sin^2(2\pi u)\,du=\left[u-\frac{\sin4\pi u}{4\pi}\right]_{0.45}^{0.55}.
$$
Therefore
$$
\boxed{P=0.1-\frac{\sin(\pi/5)}{2\pi}=0.00645\simeq0.645\%.}
$$
The central interval surrounds a node, which explains why the <probability> is much smaller than its ten-percent share of the well's length.