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Question #41481

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be the locus of points $(x,y)$ defined parametrically by $x = 2\sin t$ and $y = 2\cos 2t + 2$ for $t \in \left[0, \frac{3\pi}{2}\right]$ . Find the area of the region bounded by $C$ and the x-axis.

Options

A.
$\frac{32}{3}$
B.
$\frac{16}{3}$
C.
16
D.
8

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|Parametric Derivative Endpoint Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Parametric Derivative Endpoint Trap'.

Students often blindly apply the parametric area formula $A = |\int y(t) x'(t) dt|$ without checking if the parameterization is monotonic.

Here, $x'(t) = 2\cos t$ changes sign at $t = \pi/2$ , meaning the curve retraces itself.

The blind integral yields $|\int_{0}^{3\pi/2} 8\cos^3 t dt| = 16/3$ , and integrating with absolute values gives the 'swept area' of $16$ .

By finding the locus of points first, one realizes the curve is simply the parabola $y = 4 - x^2$ covering $x \in [-2, 2]$ , and the true geometric area is $32/3$ .

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