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GATE 2026 CH – Question 41

Engineering Mathematics · Calculus: Definite and improper integrals, partial and total derivatives, maxima and minima · 2 marks · Multiple choice

Consider the following curve in polar coordinates.

$r = 2 - 2\sin\theta$

Which one of the following is the area enclosed by the curve for $0 \le \theta \le 2\pi$ ?

  1. $3\pi$
  2. $4\pi$
  3. $5\pi$
  4. $6\pi$

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Correct answer: (D) $6\pi$

Explanation

The curve is a cardioid. Its area is $A = \frac{1}{2}\int_0^{2\pi} r^2 d\theta = \frac{1}{2}\int_0^{2\pi} 4(1 - \sin\theta)^2 d\theta = 2\int_0^{2\pi}(1 - 2\sin\theta + \sin^2\theta)d\theta = 2(2\pi - 0 + \pi) = 6\pi$.