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Question #41538

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the region in the first quadrant bounded by the hyperbola $y^2 - x^2 = 1$ , its asymptote $y = x$ , and the $y$ -axis. If the region $R$ is revolved about the line $y = x$ , the volume of the resulting solid is $V$ . Find $V$ .

Options

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

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

Axis of Integration Selection|Hyperbola Conjugate Axes|Asymptote Geometry|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to select the correct axis of integration by applying an isometric coordinate transformation to align with the hyperbola's asymptote.

The primary trap lies in the distance metric: students often fail to normalize the rotated coordinates (using $X=x+y, Y=y-x$ instead of dividing by $\sqrt{2}$ ), which leads to the incorrect volume $4\pi/3$ .

Additionally, the open interval to infinity creates a conceptual hurdle (Gabriel's Horn paradox), where the 2D area diverges but the 3D volume of revolution converges, tricking students into assuming the integral diverges or setting up the wrong improper integral.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Axis of Integration Selection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free