Question #41547

This question tests the student's ability to handle the restricted principal domains of inverse trigonometric functions when calculating areas bounded by periodic curves.

The 'Lethal Collision' occurs if a student blindly assumes $\sin^{-1}(\sin x) = x$ and $\cos^{-1}(\sin x) = \frac{\pi}{2} - x$ over the entire interval $[0, 2\pi]$, ignoring the piecewise nature of the principal branches.

This false assumption leads to integrating $|2x - \frac{\pi}{2}|$ over $[0, 2\pi]$, yielding the trap answer $\frac{25\pi^2}{8}$.

Furthermore, students might incorrectly solve for intersection points by applying the general solution of $\sin x = \cos x$ without verifying if those points lie on the correct branch of the inverse functions, leading to double-counting or missing area segments when the integration bounds cross the principal domain thresholds (e.g., at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$).