Question #40556

The problem tests the 'Condition of Tangency' to establish a linear boundary, followed by a rigorous area calculation that embeds two major traps.

The 'Extraneous Solution Rejection' trap occurs when students blindly square $x - 2 = \sqrt{x}$ to find intersection points, yielding $x = 1$ and $x = 4$.

Assuming the region is bounded between these roots, they incorrectly integrate from 1 to 4.

Simultaneously, the 'Sign Flipping Rule' trap catches students who cross-multiply the rational inequality $\frac{x - 1}{x - 4} \ge 0$ to get $x \ge 1$, which deceptively confirms the erroneous interval $[1, 4]$.

The correct logical path requires analyzing the domain of the square root and the true sign intervals, revealing the actual feasible region is $x \in [0, 1]$.