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Question #41479

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(1, 2, 3)$ , $B(2, -1, 1)$ , and $C(-1, 1, 2)$ be three points in $\mathbb{R}^3$ . The planar triangle $ABC$ is rotated by an angle of $2\pi$ about the line $L: x = y = z$ to generate a solid of revolution. If the volume of this solid is $V$ , then $V$ is equal to:

Options

A.
$\frac{16\sqrt{3}\pi}{9}$
B.
$\frac{20\sqrt{2}\pi}{3}$
C.
$\frac{32\sqrt{3}\pi}{9}$
D.
$\frac{4\sqrt{2}\pi}{9}$

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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|Area of Triangle via Determinants|Coplanarity Condition

Why This Question Is Tricky

Common Mistakes

This problem tests the generalized Pappus's Centroid Theorem for a planar region rotating about a non-coplanar skewed axis in 3D space.

A common trap is blindly applying the standard 2D Pappus's theorem ( $V = 2\pi d A$ ), which assumes the axis is coplanar with the region.

When the axis is skewed, the velocity vector of the centroid is not orthogonal to the plane's normal, requiring the scalar triple product formulation $V = 2\pi | \vec{S} \cdot (\hat{u} \times \vec{r}_c) |$ .

This forces students to integrate 3D vector dynamics with the area of a triangle via determinants, penalizing those who incorrectly assume coplanarity.

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