A parametric curve defined by $x = \cos (\frac{π u}{2}), y = \sin (\frac{π u}{2})$ in the range of 0 ≤ u ≤ 1 is rotated about the X – axis by 360 degrees. Area of the surface generated is

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PY 7: GATE ME 2017 Official Paper: Shift 1

Answer 1

Concept:

$x = \cos (\frac{π u}{2}), y = \sin (\frac{π u}{2})$

$x^{2} + y^{2} = \cos^{2} (\frac{π u}{2}) + \sin^{2} (\frac{π u}{2}) = 1$

So it represents an equation of circle in x-y plane.

Given 0 ≤ u ≤ 1

So, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

i.e., 0 ≤ θ ≤ π/2

So we get a quarter circle in x-y plane and by revolving it 360°, we get a hemisphere.

Area of hemi-sphere = 2π(1)² = 2π