The parametric curve x=( u/2), y=( u/2) for 0 u1 is rotated through 360° about the X-axis. Find the generated surface area.

GATE 2017 ME (ME1) – Question 38

Engineering Mathematics · Calculus: Partial derivatives, Taylor series, maxima and minima, Fourier series · 2 marks · Multiple choice

The parametric curve $x=\cos(\pi u/2)$, $y=\sin(\pi u/2)$ for $0\leq u\leq1$ is rotated through 360° about the X-axis. Find the generated surface area.

$\pi/2$
$\pi$
$2\pi$
$4\pi$

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Show answer and explanation

Correct answer: (C) $2\pi$

Explanation

Set $\theta=\pi u/2$, so the curve is the unit quarter-circle with $0\leq\theta\leq\pi/2$ and arc element $ds=d\theta$. Surface area of revolution is $A=\int2\pi y\,ds=2\pi\int_0^{\pi/2}\sin\theta\,d\theta=2\pi$. This is the curved surface of a unit hemisphere, option C.

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