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GATE 2024 AE – Question 37

Engineering Mathematics · Calculus: Limits, continuity, differentiability, chain rule, maxima and minima, integration · 2 marks · Multiple choice

The volume of the solid formed by a complete rotation of the shaded portion of the circle of radius $R$ about the $y$-axis is $k\pi R^3$. The value of $k$ is:

A circle of radius $R$ centred at the origin. The shaded part is the cap above a horizontal chord. The radius to the end of the chord makes $60^\circ$ with the $y$-axis.
  1. $\frac{5}{12}$
  2. $\frac{5}{24}$
  3. $\frac{7}{12}$
  4. $\frac{7}{24}$

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Correct answer: (B) $\frac{5}{24}$

Explanation

The radius to the end of the chord is $60^\circ$ from the $y$-axis, so the chord is at the height $R\cos 60^\circ = \frac{R}{2}$. Rotating the cap about the $y$-axis gives a spherical cap of height $h = R - \frac{R}{2} = \frac{R}{2}$, whose volume is $\frac{\pi h^2(3R - h)}{3} = \frac{\pi}{3}\cdot\frac{R^2}{4}\cdot\frac{5R}{2} = \frac{5}{24}\pi R^3$.