Worksheet: Arc Length of Parametric Curve

In this worksheet, we will practice finding the arc length of a parametrically defined curve.

Q1:

Calculate the arc length of over the given interval .

  • A
  • B
  • C4
  • D
  • E2

Q2:

Find the length of the curve with parametric equations 𝑥 = 3 𝑡 3 𝑡 c o s c o s and 𝑦 = 3 𝑡 3 𝑡 s i n s i n , where 0 𝑡 𝜋 .

Q3:

Express the length of the curve with parametric equations 𝑥 = 𝑡 2 𝑡 s i n and 𝑦 = 1 2 𝑡 c o s , where 0 𝑡 4 𝜋 , as an integral.

  • A 1 + 2 ( 𝑡 𝑡 ) 𝑡 4 𝜋 0 s i n c o s d
  • B 5 + 𝑡 4 𝑡 4 𝑡 𝑡 𝑡 4 𝜋 0 2 c o s s i n d
  • C 5 + 4 𝑡 𝑡 4 𝜋 0 c o s d
  • D 5 4 𝑡 𝑡 4 𝜋 0 c o s d
  • E ( 1 + 2 ( 𝑡 𝑡 ) ) 𝑡 4 𝜋 0 s i n c o s d

Q4:

Find the length of the curve with parametric equations 𝑥 = 2 𝑡 s i n 1 and 𝑦 = 1 𝑡 l n 2 , where 0 𝑡 1 2 .

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

Q5:

Find the length of the curve with parametric equations 𝑥 = 𝑡 𝑡 s i n and 𝑦 = 𝑡 𝑡 c o s , where 0 𝑡 1 .

  • A 1 2 2 + 1 2 2 1 l n
  • B 1 2 2 1 2 1 + 2 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

Q6:

Express the length of the curve with parametric equations 𝑥 = 𝑡 + 𝑒 𝑡 and 𝑦 = 𝑡 𝑒 𝑡 , where 0 𝑡 2 , as an integral.

  • A 2 𝑡 2 0 d
  • B 2 + 2 𝑒 𝑡 2 0 2 𝑡 d
  • C 2 𝑡 + 2 𝑒 𝑡 2 0 2 2 𝑡 d
  • D 2 + 2 𝑒 2 𝑡 2 0 2 𝑡 d
  • E 2 + 2 𝑒 𝑡 2 0 𝑡 2 d

Q7:

Find the length of the curve with parametric equations 𝑥 = 𝑒 𝑡 𝑡 and 𝑦 = 4 𝑒 𝑡 2 , where 0 𝑡 2 .

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

Q8:

Express the length of the curve with parametric equations 𝑥 = 𝑡 + 𝑡 and 𝑦 = 𝑡 𝑡 , where 0 𝑡 1 , as an integral.

  • A 2 𝑡 1 0 d
  • B 2 𝑡 + 2 𝑡 𝑡 1 0 2 d
  • C 2 + 2 𝑡 𝑡 1 0 d
  • D 2 + 1 2 𝑡 𝑡 1 0 d
  • E 2 𝑡 𝑡 1 0 1 4 d

Q9:

The position of a particle at time 𝑡 is 𝑡 , 𝑡 s i n c o s 2 2 . Find the distance the particle travels between 𝑡 = 0 and 𝑡 = 3 𝜋 .

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

Q10:

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

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

Q11:

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

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

Q12:

Find the length of the astroid with parametric equations 𝑥 = 𝑎 𝜃 c o s 3 and 𝑦 = 𝑎 𝜃 s i n 3 , where 𝑎 > 0 .

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

Q13:

Find the length of one arch of the cycloid with parametric equations 𝑥 = 𝑟 ( 𝑡 𝑡 ) s i n and 𝑦 = 𝑟 ( 1 𝑡 ) c o s .

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

Q14:

Consider the parametric equations 𝑥 = 𝑎 𝜃 c o s and 𝑦 = 𝑎 𝜃 s i n for 0 𝜃 2 𝜋 .

Express the arclength of this curve as an integral.

  • A 𝑎 𝜃 2 𝜋 0 2 d
  • B 1 2 𝑎 𝜃 2 𝜋 0 d
  • C 𝑎 𝜃 𝜃 𝜃 2 𝜋 0 2 2 c o s s i n d
  • D 𝑎 𝜃 2 𝜋 0 d
  • E 𝑎 𝜃 𝜃 2 𝜋 0 s i n d

Evaluate the integral.

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

Q15:

Express the length of the curve with parametric equations 𝑥 = 𝑡 𝑡 2 and 𝑦 = 𝑡 4 , where 1 𝑡 4 , as an integral.

  • A 4 𝑡 + 2 𝑡 1 𝑡 4 1 3 d
  • B 1 6 𝑡 + 4 𝑡 4 𝑡 + 1 𝑡 4 1 6 2 d
  • C 𝑡 + 𝑡 2 𝑡 + 𝑡 𝑡 4 1 8 4 3 2 d
  • D 1 6 𝑡 + 4 𝑡 4 𝑡 + 1 𝑡 4 1 6 2 d
  • E 4 𝑡 + 2 𝑡 1 𝑡 4 1 3 d

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