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.

  • A1+2(𝑡𝑡)𝑡sincosd
  • B(1+2(𝑡𝑡))𝑡sincosd
  • C5+4𝑡𝑡cosd
  • D5+𝑡4𝑡4𝑡𝑡𝑡cossind
  • E54𝑡𝑡cosd

Q3:

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

  • A5𝜋6
  • Bln94
  • C22ln
  • Dln3
  • Eln31

Q4:

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

  • A122121+2ln
  • B1221221ln
  • C122+1221ln
  • D122+121+2ln
  • E43

Q5:

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

  • A2+2𝑒𝑡d
  • B2+2𝑒2𝑡d
  • C2+2𝑒𝑡d
  • D2𝑡d
  • E2𝑡+2𝑒𝑡d

Q6:

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

  • A𝑒+1
  • B𝑒+4𝑒7
  • C𝑒
  • D𝑒2+2𝑒12
  • E12𝑒2𝑒+32

Q7:

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

  • A2𝑡𝑡d
  • B2+12𝑡𝑡d
  • C2𝑡+2𝑡𝑡d
  • D2+2𝑡𝑡d
  • E2𝑡d

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
  • B2
  • C1
  • D122
  • E62

Q9:

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

  • A422
  • B965
  • C5
  • D12𝜋
  • E13132712

Q10:

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

  • A43
  • B5583
  • C2315
  • D13
  • E105163

Q11:

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

  • A3𝑎
  • B9𝜋16𝑎
  • C9𝜋4𝑎
  • D32𝑎
  • E6𝑎

Q12:

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

  • A42𝑟
  • B4𝑟
  • C𝑟
  • D8𝑟
  • E2𝑟

Q13:

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

Express the arclength of this curve as an integral.

  • A𝑎𝜃𝜃sind
  • B𝑎𝜃𝜃𝜃cossind
  • C𝑎𝜃d
  • D12𝑎𝜃d
  • E𝑎𝜃d

Evaluate the integral.

  • A2𝜋𝑎
  • B𝜋𝑎
  • C2𝜋
  • D2𝜋𝑎
  • E𝜋𝑎

Q14:

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

  • A4𝑡+2𝑡1𝑡d
  • B4𝑡+2𝑡1𝑡d
  • C16𝑡+4𝑡4𝑡+1𝑡d
  • D𝑡+𝑡2𝑡+𝑡𝑡d
  • E16𝑡+4𝑡4𝑡+1𝑡d

Q15:

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

  • A𝜋4
  • B𝜋
  • C4𝜋
  • D𝜋2
  • E2𝜋

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