Cycloid is formed by x = a(θ−cosθ) and y = a(1−cosθ). The surface area of the curved plane obtained from the rotation of the cycloid around x axis is
Options:
(A) 16πa²/3
(B) 32πa²/3
(C) 64πa²/3
(D) 128πa²/3
Official Answer Key:
C
Step-by-Step Solution & Explanation
For the standard cycloid arch x=a(θ−sinθ), y=a(1−cosθ) rotated about the x-axis, the classical surface-of-revolution result (via S=∫2πy·ds over one arch, 0≤θ≤2π) is S = 64πa²/3. (The x-equation as printed uses cosθ rather than sinθ; taking this as a typographical rendering of the standard cycloid, the textbook result 64πa²/3 applies.)
⚠ Verification note: The parametrisation as printed (x=a(θ−cosθ)) departs from the standard cycloid (x=a(θ−sinθ)); treated as the standard cycloid form for evaluation, consistent with the official key.
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