Worksheet: Surface of Revolution of Parametric Curves

In this worksheet, we will practice using integration to find the area of the surface of revolution of a parametrically defined curve.


Consider the parametric equations 𝑥=2𝜃cos and 𝑦=2𝜃sin, where 0𝜃𝜋.The area of the surface 𝑆 obtained by rotating this parametric curve 2𝜋 radians about the 𝑥-axis can be calculated by evaluating the integral 2𝜋𝑦𝑠d where dddddd𝑠=𝑥𝜃+𝑦𝜃𝜃.

Find d𝑠.

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

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

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


Consider the parametric equations 𝑥=2𝑡1 and 𝑦=𝑡+1, where 0𝑡2. Calculate the area of the surface obtained when the curve is rotated 2𝜋 radians about the 𝑥-axis.

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

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