Worksheet: Area Enclosed by a Parametric Curve

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In this worksheet, we will practice finding areas enclosed by parametrically defined curves.

Q1:

Consider the curve defined by the parametric equations 𝑥 = 2 𝑡 c o s and 𝑦 = 3 𝑡 s i n .

Find − 6 𝑡 𝑡 s i n d 2 .

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

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

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

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

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

Q2:

Consider the curve defined by the parametric equations 𝑥 = 𝑝 3 and 𝑦 = 4 − 𝑝 2 .

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

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

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