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.

Q1:

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.

  • A𝜃d
  • B2𝜃d
  • C2𝜃d
  • D3𝜃d
  • Ed𝜃

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

  • A8𝜋
  • B𝜋
  • C2𝜋
  • D16𝜋
  • E4𝜋

Q2:

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.

  • A45𝜋
  • B25𝜋
  • C85𝜋
  • D5𝜋
  • E165𝜋

Q3:

Determine the surface area of the solid obtained by rotating the parametric curve 𝑥=1+2𝑡 and 𝑦=12𝑡, where 0𝑡2, about the 𝑦-axis.

  • A242
  • B242𝜋
  • C24
  • D24𝜋
  • E122𝜋

Q4:

Determine the surface area of the solid obtained by rotating the parametric curve 𝑥=(𝜃)cos and 𝑦=(𝜃)sin, where 0𝜃𝜋2, about the 𝑥-axis.

  • A2𝜋
  • B6𝜋5
  • C3𝜋5
  • D35
  • E65

Q5:

Determine the surface area of the solid obtained by rotating the parametric curve 𝑥=1+2𝑡 and 𝑦=4𝑡, where 0𝑡2 about the 𝑥-axis. Approximate your answer to the nearest decimal place.

Q6:

Calculate the surface area of the solid obtained by revolving the curve given by the parametric equations 𝑥=𝑒 and 𝑦=3𝑒 such that 0𝑡1 about the line 𝑥=7.5. Round your answer to two decimal places.

Q7:

Calculate the surface area of the solid obtained by revolving the curve given by the parametric equations 𝑥=𝑡 and 𝑦=𝑡+1 such that 0𝑡1 about the 𝑦-axis. Round your answer to two decimal places.

Q8:

Calculate the surface area of the solid obtained by revolving the curve given by the parametric equations 𝑥=𝑒 and 𝑦=𝑒 such that 0𝑡1 about the 𝑥-axis. Round your answer to two decimal places.

Q9:

Calculate the surface area of the solid obtained by revolving the curve given by the parametric equations 𝑥=2(𝑡)cos and 𝑦=3(𝑡)sin such that 0𝑡𝜋 about the 𝑥-axis. Round your answer to two decimal places.

Q10:

Calculate the surface area of the solid obtained by revolving the curve given by the parametric equations 𝑥=2𝑡 and 𝑦=𝑡 such that 0𝑡1 about the line 𝑦=1. Round your answer to two decimal places.

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