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Worksheet: Arc Length of Parametric Curve

Q1:

Calculate the arc length of 𝑓 ( 𝑑 ) = ο€» ο€Ή 𝑑 + 1  𝑑 , ο€Ή 𝑑 + 1  𝑑 , 2 √ 2 𝑑  2 2 c o s s i n over the given interval [ 0 , 1 ] .

  • A 5 2
  • B 7 2
  • C4
  • D 1 0 3
  • E2

Q2:

Find the length of the curve with parametric equations π‘₯ = 3 𝑑 βˆ’ 3 𝑑 c o s c o s and 𝑦 = 3 𝑑 βˆ’ 3 𝑑 s i n s i n , where 0 ≀ 𝑑 ≀ πœ‹ .

Q3:

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

  • A ο„Έ √ 1 + 2 ( 𝑑 βˆ’ 𝑑 ) 𝑑 4 πœ‹ 0 s i n c o s d
  • B ο„Έ √ 5 + 𝑑 βˆ’ 4 𝑑 βˆ’ 4 𝑑 𝑑 𝑑 4 πœ‹ 0 2 c o s s i n d
  • C ο„Έ √ 5 + 4 𝑑 𝑑 4 πœ‹ 0 c o s d
  • D ο„Έ √ 5 βˆ’ 4 𝑑 𝑑 4 πœ‹ 0 c o s d
  • E ο„Έ ( 1 + 2 ( 𝑑 βˆ’ 𝑑 ) ) 𝑑 4 πœ‹ 0 s i n c o s d

Q4:

Find the length of the curve with parametric equations π‘₯ = 2 𝑑 s i n βˆ’ 1 and 𝑦 = ο€Ή 1 βˆ’ 𝑑  l n 2 , where 0 ≀ 𝑑 ≀ 1 2 .

  • A √ 5 πœ‹ 6
  • B l n 3 βˆ’ 1
  • C l n ο€Ό 9 4 
  • D l n 3
  • E 2 2 l n

Q5:

Find the length of the curve with parametric equations π‘₯ = 𝑑 𝑑 s i n and 𝑦 = 𝑑 𝑑 c o s , where 0 ≀ 𝑑 ≀ 1 .

  • A 1 2 √ 2 + 1 2 ο€» √ 2 βˆ’ 1  l n
  • B βˆ’ 1 2 √ 2 βˆ’ 1 2 ο€» 1 + √ 2  l n
  • C βˆ’ 1 2 √ 2 βˆ’ 1 2 ο€» √ 2 βˆ’ 1  l n
  • D 1 2 √ 2 + 1 2 ο€» 1 + √ 2  l n
  • E 4 3

Q6:

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

  • A ο„Έ √ 2 𝑑 2 0 d
  • B ο„Έ ο€Ή 2 + 2 𝑒  𝑑 2 0 βˆ’ 2 𝑑 d
  • C ο„Έ √ 2 𝑑 + 2 𝑒 𝑑 2 0 2 βˆ’ 2 𝑑 d
  • D ο„Έ √ 2 + 2 𝑒 2 𝑑 2 0 βˆ’ 2 𝑑 d
  • E ο„Έ √ 2 + 2 𝑒 𝑑 2 0 𝑑 2 d

Q7:

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

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

Q8:

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

  • A ο„Έ √ 2 𝑑 1 0 d
  • B ο„Έ √ 2 𝑑 + 2 𝑑 𝑑 1 0 2 d
  • C ο„Έ ο„ž 2 + 2 𝑑 𝑑 1 0 d
  • D ο„Έ ο„ž 2 + 1 2 𝑑 𝑑 1 0 d
  • E ο„Έ √ 2 𝑑 𝑑 1 0 βˆ’ 1 4 d

Q9:

The position of a particle at time 𝑑 is ο€Ί 𝑑 , 𝑑  s i n c o s 2 2 . Find the distance the particle travels between 𝑑 = 0 and 𝑑 = 3 πœ‹ .

  • A 1 2 √ 2
  • B √ 2
  • C6
  • D 6 √ 2
  • E1

Q10:

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

  • A 1 3 √ 1 3 βˆ’ 2 7 1 2
  • B 9 6 5
  • C5
  • D 4 √ 2 βˆ’ 2
  • E 1 2 πœ‹

Q11:

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

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

Q12:

Find the length of the astroid with parametric equations π‘₯ = π‘Ž πœƒ c o s 3 and 𝑦 = π‘Ž πœƒ s i n 3 , where π‘Ž > 0 .

  • A 3 π‘Ž
  • B 3 2 π‘Ž
  • C 9 πœ‹ 4 π‘Ž 2
  • D 6 π‘Ž
  • E 9 πœ‹ 1 6 π‘Ž 2

Q13:

Find the length of one arch of the cycloid with parametric equations π‘₯ = π‘Ÿ ( 𝑑 βˆ’ 𝑑 ) s i n and 𝑦 = π‘Ÿ ( 1 βˆ’ 𝑑 ) c o s .

  • A 4 √ 2 π‘Ÿ
  • B 4 π‘Ÿ
  • C 2 π‘Ÿ
  • D 8 π‘Ÿ
  • E π‘Ÿ

Q14:

Consider the parametric equations π‘₯ = π‘Ž πœƒ c o s and 𝑦 = π‘Ž πœƒ s i n for 0 ≀ πœƒ ≀ 2 πœ‹ .

Express the arclength of this curve as an integral.

  • A ο„Έ π‘Ž πœƒ 2 πœ‹ 0 2 d
  • B 1 2 ο„Έ π‘Ž πœƒ 2 πœ‹ 0 d
  • C ο„Έ π‘Ž √ πœƒ βˆ’ πœƒ πœƒ 2 πœ‹ 0 2 2 c o s s i n d
  • D ο„Έ π‘Ž πœƒ 2 πœ‹ 0 d
  • E ο„Έ π‘Ž πœƒ πœƒ 2 πœ‹ 0 s i n d

Evaluate the integral.

  • A 2 πœ‹ π‘Ž
  • B πœ‹ π‘Ž 2
  • C 2 πœ‹ π‘Ž 2
  • D πœ‹ π‘Ž
  • E 2 πœ‹

Q15:

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

  • A ο„Έ √ 4 𝑑 + 2 𝑑 βˆ’ 1 𝑑 4 1 3 d
  • B ο„Έ ο€Ή 1 6 𝑑 + 4 𝑑 βˆ’ 4 𝑑 + 1  𝑑 4 1 6 2 d
  • C ο„Έ √ 𝑑 + 𝑑 βˆ’ 2 𝑑 + 𝑑 𝑑 4 1 8 4 3 2 d
  • D ο„Έ √ 1 6 𝑑 + 4 𝑑 βˆ’ 4 𝑑 + 1 𝑑 4 1 6 2 d
  • E ο„Έ ο€Ή 4 𝑑 + 2 𝑑 βˆ’ 1  𝑑 4 1 3 d