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Lesson Worksheet: Arc Length of Parametric Curves Mathematics • Higher Education

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 π‘₯=3π‘‘βˆ’3𝑑coscos and 𝑦=3π‘‘βˆ’3𝑑sinsin, where 0β‰€π‘‘β‰€πœ‹.

Q2:

Express the length of the curve with parametric equations π‘₯=π‘‘βˆ’2𝑑sin and 𝑦=1βˆ’2𝑑cos, where 0≀𝑑≀4πœ‹, as an integral.

  • Aο„Έβˆš1+2(π‘‘βˆ’π‘‘)𝑑οŠͺοŽ„οŠ¦sincosd
  • Bο„Έ(1+2(π‘‘βˆ’π‘‘))𝑑οŠͺοŽ„οŠ¦sincosd
  • Cο„Έβˆš5+4𝑑𝑑οŠͺοŽ„οŠ¦cosd
  • Dο„Έβˆš5+π‘‘βˆ’4π‘‘βˆ’4𝑑𝑑𝑑οŠͺοŽ„οŠ¦οŠ¨cossind
  • Eο„Έβˆš5βˆ’4𝑑𝑑οŠͺοŽ„οŠ¦cosd

Q3:

Find the length of the curve with parametric equations π‘₯=2𝑑sin and 𝑦=ο€Ή1βˆ’π‘‘ο…ln, where 0≀𝑑≀12.

  • A√5πœ‹6
  • Blnο€Ό94
  • C22ln
  • Dln3
  • Eln3βˆ’1

Q4:

Find the length of the curve with parametric equations π‘₯=𝑑𝑑sin and 𝑦=𝑑𝑑cos, where 0≀𝑑≀1.

  • Aβˆ’12√2βˆ’12ο€»1+√2ln
  • Bβˆ’12√2βˆ’12ο€»βˆš2βˆ’1ln
  • C12√2+12ο€»βˆš2βˆ’1ln
  • D12√2+12ο€»1+√2ln
  • E43

Q5:

Express the length of the curve with parametric equations π‘₯=𝑑+π‘’οŠ±ο and 𝑦=π‘‘βˆ’π‘’οŠ±ο, where 0≀𝑑≀2, as an integral.

  • Aο„Έο€Ή2+2π‘’ο…π‘‘οŠ¨οŠ¦οŠ±οŠ¨οd
  • Bο„Έβˆš2+2𝑒2π‘‘οŠ¨οŠ¦οŠ±οŠ¨οd
  • Cο„Έβˆš2+2π‘’π‘‘οŠ¨οŠ¦οοŽ‘d
  • Dο„Έβˆš2π‘‘οŠ¨οŠ¦d
  • Eο„Έβˆš2𝑑+2π‘’π‘‘οŠ¨οŠ¦οŠ¨οŠ±οŠ¨οd

Q6:

Find the length of the curve with parametric equations π‘₯=π‘’βˆ’π‘‘ο and 𝑦=4π‘’ο‘‰οŽ‘, where 0≀𝑑≀2.

  • A𝑒+1
  • B𝑒+4π‘’βˆ’7
  • C𝑒
  • D𝑒2+2π‘’βˆ’12οŠͺ
  • E12π‘’βˆ’2𝑒+32

Q7:

Express the length of the curve with parametric equations π‘₯=𝑑+βˆšπ‘‘ and 𝑦=π‘‘βˆ’βˆšπ‘‘, where 0≀𝑑≀1, as an integral.

  • Aο„Έβˆš2π‘‘π‘‘οŠ§οŠ¦οŠ±οŽ οŽ£d
  • Bο„Έο„ž2+12π‘‘π‘‘οŠ§οŠ¦d
  • Cο„Έβˆš2𝑑+2π‘‘π‘‘οŠ§οŠ¦οŠ¨d
  • Dο„Έο„ž2+2π‘‘π‘‘οŠ§οŠ¦d
  • Eο„Έβˆš2π‘‘οŠ§οŠ¦d

Q8:

The position of a particle at time 𝑑 is 𝑑,𝑑sincos. Find the distance the particle travels between 𝑑=0 and 𝑑=3πœ‹.

  • A6
  • B√2
  • C1
  • D12√2
  • E6√2

Q9:

Find the length of the curve with parametric equations π‘₯=1+3π‘‘οŠ¨ and 𝑦=4+2π‘‘οŠ©, where 0≀𝑑≀1.

  • A4√2βˆ’2
  • B965
  • C5
  • D12πœ‹
  • E13√13βˆ’2712

Q10:

Find the length of the curve with parametric equations π‘₯=π‘‘οŠ¨ and 𝑦=13π‘‘οŠ©, where 0≀𝑑≀1.

  • A43
  • B5√5βˆ’83
  • C2315
  • D13
  • E10√5βˆ’163

This lesson includes 6 additional questions for subscribers.

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