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Worksheet: Surface of Revolution of Parametric Curves

Q1:

Consider the parametric equations π‘₯ = 2 πœƒ c o s and 𝑦 = 2 πœƒ s i n , where 0 ≀ πœƒ ≀ πœ‹ .The area of the surface 𝑆 obtained by rotating this parametric curve 2 πœ‹ radians about the π‘₯ -axis can be calculated by evaluating the integral ο„Έ 2 πœ‹ 𝑦 𝑠 d where d d d d d d 𝑠 = ο„Ÿ ο€½ π‘₯ πœƒ  + ο€½ 𝑦 πœƒ  πœƒ 2 2 .

Find d 𝑠 .

  • A d πœƒ
  • B βˆ’ 2 πœƒ d
  • C βˆ’ πœƒ d
  • D 2 πœƒ d
  • E 3 πœƒ d

Hence, find the surface area of 𝑆 by evaluating the integral.

  • A 1 6 πœ‹
  • B 4 πœ‹
  • C 2 πœ‹
  • D 8 πœ‹
  • E πœ‹

Q2:

Consider the parametric equations π‘₯ = 2 𝑑 βˆ’ 1 and 𝑦 = 𝑑 + 1 , where 0 ≀ 𝑑 ≀ 2 . Calculate the area of the surface obtained when the curve is rotated 2 πœ‹ radians about the π‘₯ -axis.

  • A 4 √ 5 πœ‹
  • B √ 5 πœ‹
  • C 1 6 √ 5 πœ‹
  • D 8 √ 5 πœ‹
  • E 2 √ 5 πœ‹