\([y]\) represents the greatest integer less than or equal to \(y\) and \(\{y\}\) represents the fractional part of \(y\). If \[ \lim_{x \to 0^{+}} \left( [1-x] + \frac{a^{2[1-x] + \{1-x\} - 1}}{2[1-x] + \{1-x\}^{2}} \right) = 11, \] then \(a =\) ?
The area of the region \( \{(x, y): 0 \leq y \leq x^2 + 1, \, 0 \leq y \leq x + 1, \, 0 \leq x \leq 2\ \) is:}