Worksheet: Arc Length of Parametric Curves

In this worksheet, we will practice using integration to find the arc length of a parametrically defined curve.

Q1:

Find the length of the curve with parametric equations 𝑥=3𝑡3𝑡coscos and 𝑦=3𝑡3𝑡sinsin, where 0𝑡𝜋.

Q2:

Express the length of the curve with parametric equations 𝑥=𝑡2𝑡sin and 𝑦=12𝑡cos, where 0𝑡4𝜋, as an integral.

  • A 1 + 2 ( 𝑡 𝑡 ) 𝑡 s i n c o s d
  • B ( 1 + 2 ( 𝑡 𝑡 ) ) 𝑡 s i n c o s d
  • C 5 + 4 𝑡 𝑡 c o s d
  • D 5 + 𝑡 4 𝑡 4 𝑡 𝑡 𝑡 c o s s i n d
  • E 5 4 𝑡 𝑡 c o s d

Q3:

Find the length of the curve with parametric equations 𝑥=2𝑡sin and 𝑦=1𝑡ln, where 0𝑡12.

  • A 5 𝜋 6
  • B l n 9 4
  • C 2 2 l n
  • D l n 3
  • E l n 3 1

Q4:

Find the length of the curve with parametric equations 𝑥=𝑡𝑡sin and 𝑦=𝑡𝑡cos, where 0𝑡1.

  • A 1 2 2 1 2 1 + 2 l n
  • B 1 2 2 1 2 2 1 l n
  • C 1 2 2 + 1 2 2 1 l n
  • D 1 2 2 + 1 2 1 + 2 l n
  • E 4 3

Q5:

Express the length of the curve with parametric equations 𝑥=𝑡+𝑒 and 𝑦=𝑡𝑒, where 0𝑡2, as an integral.

  • A 2 + 2 𝑒 𝑡 d
  • B 2 + 2 𝑒 2 𝑡 d
  • C 2 + 2 𝑒 𝑡 d
  • D 2 𝑡 d
  • E 2 𝑡 + 2 𝑒 𝑡 d

Q6:

Find the length of the curve with parametric equations 𝑥=𝑒𝑡 and 𝑦=4𝑒, where 0𝑡2.

  • A 𝑒 + 1
  • B 𝑒 + 4 𝑒 7
  • C 𝑒
  • D 𝑒 2 + 2 𝑒 1 2
  • E 1 2 𝑒 2 𝑒 + 3 2

Q7:

Express the length of the curve with parametric equations 𝑥=𝑡+𝑡 and 𝑦=𝑡𝑡, where 0𝑡1, as an integral.

  • A 2 𝑡 𝑡 d
  • B 2 + 1 2 𝑡 𝑡 d
  • C 2 𝑡 + 2 𝑡 𝑡 d
  • D 2 + 2 𝑡 𝑡 d
  • E 2 𝑡 d

Q8:

The position of a particle at time 𝑡 is 𝑡,𝑡sincos. Find the distance the particle travels between 𝑡=0 and 𝑡=3𝜋.

  • A6
  • B 2
  • C1
  • D 1 2 2
  • E 6 2

Q9:

Find the length of the curve with parametric equations 𝑥=1+3𝑡 and 𝑦=4+2𝑡, where 0𝑡1.

  • A 4 2 2
  • B 9 6 5
  • C5
  • D 1 2 𝜋
  • E 1 3 1 3 2 7 1 2

Q10:

Find the length of the curve with parametric equations 𝑥=𝑡 and 𝑦=13𝑡, where 0𝑡1.

  • A 4 3
  • B 5 5 8 3
  • C 2 3 1 5
  • D 1 3
  • E 1 0 5 1 6 3

Q11:

Find the length of the astroid with parametric equations 𝑥=𝑎𝜃cos and 𝑦=𝑎𝜃sin, where 𝑎>0.

  • A 3 𝑎
  • B 9 𝜋 1 6 𝑎
  • C 9 𝜋 4 𝑎
  • D 3 2 𝑎
  • E 6 𝑎

Q12:

Find the length of one arch of the cycloid with parametric equations 𝑥=𝑟(𝑡𝑡)sin and 𝑦=𝑟(1𝑡)cos.

  • A 4 2 𝑟
  • B 4 𝑟
  • C 𝑟
  • D 8 𝑟
  • E 2 𝑟

Q13:

Consider the parametric equations 𝑥=𝑎𝜃cos and 𝑦=𝑎𝜃sin for 0𝜃2𝜋.

Express the arclength of this curve as an integral.

  • A 𝑎 𝜃 𝜃 s i n d
  • B 𝑎 𝜃 𝜃 𝜃 c o s s i n d
  • C 𝑎 𝜃 d
  • D 1 2 𝑎 𝜃 d
  • E 𝑎 𝜃 d

Evaluate the integral.

  • A 2 𝜋 𝑎
  • B 𝜋 𝑎
  • C 2 𝜋
  • D 2 𝜋 𝑎
  • E 𝜋 𝑎

Q14:

Express the length of the curve with parametric equations 𝑥=𝑡𝑡 and 𝑦=𝑡, where 1𝑡4, as an integral.

  • A 4 𝑡 + 2 𝑡 1 𝑡 d
  • B 4 𝑡 + 2 𝑡 1 𝑡 d
  • C 1 6 𝑡 + 4 𝑡 4 𝑡 + 1 𝑡 d
  • D 𝑡 + 𝑡 2 𝑡 + 𝑡 𝑡 d
  • E 1 6 𝑡 + 4 𝑡 4 𝑡 + 1 𝑡 d

Q15:

Find the arc length of the curve defined by the parametric equations 𝑥=𝑡cos and 𝑦=𝑡sin.

  • A 𝜋 4
  • B 𝜋
  • C 4 𝜋
  • D 𝜋 2
  • E 2 𝜋

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