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Question #41540

Difficulty: 8/10 | 🟢 Free

The Question

Let $I = \int_{0}^{1} \int_{x^2}^{x} \arcsin\left(\sin\left(\frac{\pi y}{x}\right)\right) dy \, dx$ . If $I = \frac{p\pi}{q}$ , where $p$ and $q$ are coprime positive integers, find the value of $p+q$ .

Options

A.
199
B.
9
C.
193
D.
201

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Horizontal vs Vertical Strips|Homogeneous Function Homogeneity|Composition Continuity|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume $\arcsin(\sin \theta) = \theta$ universally, ignoring the piecewise continuity of the composite function.

Even if identified, the non-differentiable boundary $y = x/2$ creates a 'Lethal Collision' by intersecting the integration region $x^2 \le y \le x$ .

Evaluating via standard vertical strips requires a tedious domain split at $x=1/2$ .

The elegant bypass uses the homogeneity substitution $y=xt$ followed by integration by parts (which is analytically equivalent to changing the order of integration to horizontal strips), reducing the problem to a single-variable integral and cleanly isolating the non-differentiable point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Horizontal vs Vertical Strips
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free