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Question #41541

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be a curve defined parametrically by $x = t^2, y = 2t$ for $t \in \mathbb{R}$ . Let $L$ be the polar line of the point $P(-1, 0)$ with respect to $C$ . If $R$ is the finite region bounded by $C$ and $L$ , evaluate the integral: $I = \iint_R \frac{y^2}{(x+1)^3 \sqrt{1-x}} \,dx\,dy$

Options

A.
$\frac{\pi}{\sqrt{2}}$
B.
$\frac{\pi}{2\sqrt{2}}$
C.
$\frac{3\pi}{4\sqrt{2}}$
D.
$\frac{\pi}{4\sqrt{2}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Horizontal vs Vertical Strips|Parametric Form Manipulation|Polar and Pole Duality|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge coordinate geometry and multivariable calculus by identifying a region defined through pole-polar duality.

The primary trap is the 'Denominator Restriction' at the boundary $x=1$ , which acts as a singularity.

Students who attempt to integrate using horizontal strips will face an intractable function, forcing a strategic switch to vertical strips.

Furthermore, failing to recognize the improper nature of the integral at the polar line $L$ often leads to incorrect limit evaluations.

The problem rigorously demands mastery over parametric transformations, order of integration reversal, and advanced trigonometric substitutions to resolve the singularity.

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