Worksheet: Area Enclosed by Parametric Curves

In this worksheet, we will practice using integration to find the area under a curve defined by parametric functions.

Q1:

Consider the curve defined by the parametric equations 𝑥=2𝑡cos and 𝑦=3𝑡sin.

Find 6𝑡𝑡sind.

  • A3𝑡+322𝑡+𝑐sin
  • B3𝑡+122𝑡+𝑐sin
  • C3𝑡322𝑡+𝑐sin
  • D6𝑡+322𝑡+𝑐sin
  • E3𝑡+322𝑡+𝑐sin

Find the area under the curve when 0𝑡𝜋.

  • A𝜋
  • B2𝜋
  • C3𝜋
  • D𝜋3
  • E6𝜋

Now, by taking 0𝑡2𝜋, find the total area inside the curve.

  • A6𝜋
  • B2𝜋
  • C3𝜋
  • D𝜋3
  • E𝜋

Q2:

Consider the curve defined by the parametric equations 𝑥=𝑝 and 𝑦=4𝑝.

Find the area under the curve where 0𝑝1.

Find the area under the curve where 0𝑝2.

Q3:

Determine the area trapped between the 𝑥-axis and the curve with parametric equations 𝑥=𝑡 and 𝑦=𝑒 on the interval [0,1]. Approximate your answer to the nearest one decimal place.

Q4:

Determine the area trapped between the 𝑥-axis and the parametric curve defined by the equations 𝑥=𝑡2 and 𝑦=𝑡 on the interval [0,2].

  • A323
  • B163
  • C8
  • D83
  • E16

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