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Question #40871

Difficulty: 8/10 | 🟢 Free

The Question

Consider a space curve $C$ parameterized by $\vec{r}(t) = 3\sin t \hat{i} + 2\sin 2t \hat{j} + \cos(e^t) \hat{k}$ for $t \in [0, 2\pi]$ . Let $C'$ be the orthogonal projection of $C$ onto the $xy$ -plane. What is the total geometric area of the region(s) enclosed by $C'$ ?

Options

A.
0
B.
8
C.
16
D.
32

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

Coordinate Axis Projections|Parametric Curve Area|Region Ambiguity

Why This Question Is Tricky

Common Mistakes

This question tests the 'Region Ambiguity' trap in parametric area integration.

Students often apply the standard area integral $\frac{1}{2} \oint (x dy - y dx)$ directly over the entire given parameter domain $[0, 2\pi]$ .

Because the projected curve (a lemniscate of Gerono variant) traces two loops with opposite orientations, the net integral evaluates to 0.

To find the true geometric area, students must identify the self-intersection at the origin, partition the domain into $[0, \pi]$ and $[\pi, 2\pi]$ , and sum the absolute values of the integrals for each bounded region.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Coordinate Axis Projections
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free