Worksheet: Surface of Revolution of Parametric Curves

In this worksheet, we will practice finding the area of the surface of revolution for a parametrized curve.

Q1:

Consider the parametric equations 𝑥 = 2 c o s 𝜃 and 𝑦 = 2 s i n 𝜃 , where 0 𝜃 𝜋 .The area of the surface 𝑆 obtained by rotating this parametric curve 2 𝜋 radians about the 𝑥 -axis can be calculated by evaluating the integral 2 𝜋 𝑦 d 𝑠 where d 𝑠 = d 𝑥 d 𝜃 + d 𝑦 d 𝜃 d 𝜃 .

Find d 𝑠 .

  • A d 𝜃
  • B 2 d 𝜃
  • C d 𝜃
  • D 2 d 𝜃
  • E 3 d 𝜃

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

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

Q2:

Consider the parametric equations 𝑥 = 2 𝑡 1 and 𝑦 = 𝑡 + 1 , where 0 𝑡 2 . Calculate the area of the surface obtained when the curve is rotated 2 𝜋 radians about the 𝑥 -axis.

  • A 4 5 𝜋
  • B 5 𝜋
  • C 1 6 5 𝜋
  • D 8 5 𝜋
  • E 2 5 𝜋

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