Worksheet: Area Enclosed by Parametric Curves

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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.

  • A 3 𝑡 + 3 2 2 𝑡 + 𝑐 s i n
  • B − 3 𝑡 + 1 2 2 𝑡 + 𝑐 s i n
  • C − 3 𝑡 − 3 2 2 𝑡 + 𝑐 s i n
  • D − 6 𝑡 + 3 2 2 𝑡 + 𝑐 s i n
  • E − 3 𝑡 + 3 2 2 𝑡 + 𝑐 s i n

Find the area under the curve when 0≤𝑡≤𝜋.

  • A 𝜋
  • B 2 𝜋
  • C 3 𝜋
  • D 𝜋 3
  • E 6 𝜋

Now, by taking 0≤𝑡≤2𝜋, find the total area inside the curve.

  • A 6 𝜋
  • B 2 𝜋
  • C 3 𝜋
  • D 𝜋 3
  • E 𝜋

Q2:

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

Find the area under the curve where 0≤𝑝≤1.

Find the area under the curve where 0≤𝑝≤2.

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