Lesson Worksheet: Area Enclosed by Parametric Curves Mathematics

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

Question 1

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

Question 2

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.

Question 3

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.

Question 4

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

Question 5

Determine the area inside the curve defined by the parametric equations 𝑥=2(2𝑡)cos and 𝑦=2(2𝑡)sin.

  • A2𝜋
  • B𝜋
  • C2𝜋
  • D4𝜋
  • E𝜋

Question 6

Find the area between the following two curves: curve 1 that is defined by the parametric equations 𝑥=2𝑡, 𝑦=2𝑡 and curve 2 that is defined by the parametric equations 𝑥=2𝑡, 𝑦=2𝑡 on the interval 0𝑥2.

  • A323
  • B503
  • C1216
  • D23
  • E43

Question 7

Find the area between curve 1 defined by the parametric equations 𝑥=2𝑡cos, 𝑦=2𝑡sin and curve 2 defined by the parametric equations 𝑥=2𝑡cos, 𝑦=3𝑡sin on the interval 0𝑥2.

  • A𝜋2
  • B3𝜋2
  • C3𝜋2
  • D𝜋2
  • E𝜋

Question 8

Find the area between the following curves: curve 1 that is defined by the parametric equations 𝑥=𝑡, 𝑦=2𝑡; curve 2 that is defined by the parametric equations 𝑥=2, 𝑦>0; and curve 3 that is defined by the parametric equation 𝑥>0, 𝑦=0.

  • A103
  • B23
  • C23
  • D223
  • E83

Question 9

Consider the curve defined by the parametric equations 𝑥=𝑡, 𝑦=𝑡sin, 𝑡0. Find the area between the curve and the 𝑥-axis on the interval 0𝑥1. Approximate your answer to three decimal places.

Question 10

Consider the curve defined by the parametric equations 𝑥=12𝑡cos, 𝑦=2𝑡sin. Find the area between the curve and the 𝑥-axis on the interval 0𝑥12.

  • A2𝜋
  • B2𝜋
  • C𝜋
  • D𝜋2
  • E6𝜋

This lesson includes 72 additional question variations for subscribers.

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