Question #40534

This question tests 'Region Ambiguity' by subverting expectations around even multiplicity roots and modulus functions.

The function contains the factor $(x-1)^2$, which typically implies the curve touches the $x$-axis without crossing it.

However, the modulus term $(1 - |x-2|)$ also contributes a root at $x=1$, effectively making the overall multiplicity odd at $x=1$.

This causes the curve to cross the axis, creating a hidden bounded region below the $x$-axis for $x \in [0,1]$.

Students who blindly assume even multiplicity implies no crossing will miss this region (Trap: 374) or fail to take its absolute value (Trap: 373).

Furthermore, ignoring the non-differentiable point at $x=2$ and integrating without splitting the modulus leads to a completely incorrect area (Trap: 405).