Let r and be the polar coordinates defined by x = r and y = r. The area of the cardioid r = a(1 - ), 0 2, is

GATE 2024 CH – Question 47

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

Let $r$ and $\theta$ be the polar coordinates defined by $x = r\cos\theta$ and $y = r\sin\theta$. The area of the cardioid $r = a(1 - \cos\theta)$, $0 \le \theta \le 2\pi$, is

a cardioid with its cusp at the origin, opening to the right.

$\frac{3\pi a^2}{2}$
$\frac{2\pi a^2}{3}$
$3\pi a^2$
$2\pi a^2$

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Correct answer: (A) $\frac{3\pi a^2}{2}$

Explanation

$A = \frac{1}{2}\int_0^{2\pi}r^2d\theta = \frac{a^2}{2}\int_0^{2\pi}(1 - 2\cos\theta + \cos^2\theta)d\theta = \frac{a^2}{2}(2\pi - 0 + \pi) = \frac{3\pi a^2}{2}$.