Worksheet: Arc Length of a Polar Curve

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In this worksheet, we will practice finding the length of a curve defined by polar equations using integration.

Q1:

Write the integral for the arc length of the spiral 𝑟 = 𝜃 between 𝜃 = 0 and 𝜃 = 𝜋 . Do not evaluate the integral.

  • A √ 1 + 𝑒 𝑑 𝜃
  • B √ 1 + 𝜃 𝑑 𝜃
  • C √ 1 − 𝜃 𝑑 𝜃
  • D √ 1 − 𝑒 𝑑 𝜃

Q2:

The purpose of this question is to get improved estimates on the length of a spiral curve.

Use the fact that 𝑥 < 1 + 𝑥 < ( 1 + 𝑥 ) when 𝑥 > 0 to find lower and upper bounds for the length 𝐿 of the spiral 𝑟 = 𝜃 between 𝜃 = 0 and 𝜃 = 𝜋 . Give your answer to 4 decimal places.

  • A 4 . 9 3 4 8 < 𝐿 < 6 . 5 0 5 6
  • B 4 . 9 3 4 8 < 𝐿 < 1 0 . 8 6 9 6
  • C 9 . 8 6 9 6 < 𝐿 < 1 7 . 1 5 2 7
  • D 8 . 0 7 6 4 < 𝐿 < 1 0 . 8 6 9 6
  • E 4 . 9 3 4 8 < 𝐿 < 8 . 0 7 6 4

By comparing √ 1 + 𝑥 to the average of 𝑥 and 1 + 𝑥 when 𝑥 > 0 , find better bounds for estimating 𝐿 . Give your answer to 4 decimal places.

  • A 6 . 5 0 5 6 < 𝐿 < 1 0 . 8 6 9 6
  • B 4 . 9 3 4 8 < 𝐿 < 6 . 5 0 5 6
  • C 6 . 5 0 5 6 < 𝐿 < 8 . 0 7 6 4
  • D 4 . 9 3 4 8 < 𝐿 < 8 . 0 7 6 4
  • E 4 . 9 3 4 8 < 𝐿 < 1 0 . 8 6 9 6

Q3:

Let 𝑠 be the arc length of the polar curve 𝑟 = 3 𝜃 over the interval 0 ≤ 𝜃 ≤ 𝜋 2 . Express 𝑠 as a definite integral.

  • A 𝑠 = 3 √ 𝜃 + 1 𝜃 d
  • B 𝑠 = √ 3 𝜃 + 3 𝜃 d
  • C 𝑠 = 4 𝜃 d
  • D 𝑠 = √ 1 0 𝜃 d
  • E 𝑠 = √ 3 𝜃 + 3 𝜃 d

Using a calculator, or otherwise, find the value of 𝑠 giving your answer to 4 decimal places.

Q4:

Find the total arc length of 𝑟 = 3 𝜃 s i n .

  • A6
  • B3
  • C 6 𝜋
  • D 3 𝜋
  • E 9 𝜋

Q5:

Find the arc length of the polar curve 𝑟 = 𝜃 + 𝜃 s i n c o s , where 𝜃 lies in the interval [ 0 , 𝜋 ] .

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

Q6:

Consider the polar curve 𝑟 = 1 𝜃 , where 𝜃 lies in the interval [ 0 , 2 𝜋 ] . Find a definite integral that represents the arc length of this curve.

  • A √ 𝜃 − 1 𝜃 𝜃 d
  • B √ 𝜃 + 1 𝜃 𝜃 d
  • C 𝜃 + 1 𝜃 𝜃 d
  • D √ 𝜃 − 1 𝜃 𝜃 d
  • E0

Q7:

Consider the polar curve 𝑟 = 1 + 𝜃 s i n , where 𝜃 lies in the interval [ 0 , 2 𝜋 ] . Find a definite integral that represents the arc length of this curve.

  • A √ 1 + 𝜃 + 𝜃 𝜃 s i n c o s d
  • B ( 2 + 2 𝜃 ) 𝜃 s i n d
  • C √ 1 + 𝜃 − 𝜃 𝜃 s i n c o s d
  • D ( 1 + 𝜃 + 𝜃 ) 𝜃 s i n c o s d
  • E √ 2 + 2 𝜃 𝜃 s i n d

Q8:

Find the arc length of the polar curve 𝑟 = 5 , where 𝜃 lies in the interval [ 0 , 2 𝜋 ] .

  • A 2 5 √ 1 + ( 5 ) 5 l n l n
  • B √ 1 + ( 5 ) ( 2 5 − 1 ) l n
  • C √ 1 + ( 5 ) 5 ( 2 5 − 1 ) l n l n
  • D √ 2 ( 2 5 − 1 )
  • E 2 √ 1 + ( 5 ) 5 ( 2 5 − 1 ) l n l n

Q9:

Find a definite integral that represents the arc length of 𝑟 = 1 + 𝜃 s i n on the interval 0 ≤ 𝜃 ≤ 2 𝜋 .

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

Q10:

Find the arc length of the polar curve given by 𝑟 = 𝑒 on the interval 0 ≤ 𝜃 ≤ 2 .

  • A √ 1 0 𝑒 − 1
  • B 4 3 𝑒 − 1
  • C 1 0 𝑒 − 1
  • D √ 1 0 3 𝑒 − 1
  • E √ 1 0 3 𝑒 − 1

Q11:

Find the arc length of the polar curve 𝑟 = 1 − 𝜃 s i n over the interval 0 ≤ 𝜃 ≤ 2 𝜋 .

Q12:

Find the arc length of the polar curve 𝑟 = 6 , where 𝜃 lies in the interval 0 , 𝜋 2 .

  • A √ 6 𝜋 2
  • B 1 8 𝜋
  • C 6 𝜋
  • D 3 𝜋 2
  • E 3 𝜋

Q13:

Consider the polar curve 𝑟 = 4 𝜃 c o s , where 𝜃 lies in the interval 0 , 𝜋 2 . Find a definite integral that represents the arc length of this curve.

  • A 4 𝜃 d
  • B 2 √ 𝜃 − 𝜃 𝜃 c o s s i n d
  • C ( 4 𝜃 − 4 𝜃 ) 𝜃 c o s s i n d
  • D 1 6 𝜃 d
  • E 2 √ 𝜃 + 𝜃 𝜃 c o s s i n d

Q14:

Find the arc length of the polar curve 𝑟 = 6 𝜃 c o s , where 𝜃 lies in the interval 0 , 𝜋 2 .

  • A6
  • B 6 𝜋
  • C 3 𝜋
  • D 1 8 𝜋
  • E 3 2 𝜋

Q15:

Find the arc length of the cardioid 𝑟 = 2 + 2 𝜃 c o s .

Q16:

Consider the polar curve 𝑟 = 2 𝜃 s e c , where 𝜃 lies in the interval 0 , 𝜋 3 . Find a definite integral that represents the arc length of this curve.

  • A 2 𝜃 𝜃 s e c d
  • B 4 𝜃 𝜃 s e c d
  • C √ 4 𝜃 + 2 𝜃 𝜃 𝜃 s e c s e c t a n d
  • D √ 2 𝜃 + 2 𝜃 𝜃 𝜃 s e c s e c t a n d
  • E √ 2 𝜃 − 2 𝜃 𝜃 𝜃 s e c s e c t a n d

Q17:

Find the arc length of the polar curve 𝑟 = 8 + 8 𝜃 c o s over the interval 0 ≤ 𝜃 ≤ 𝜋 .

Q18:

Let 𝑠 be the arc length of the polar curve 𝑟 = 2 𝜃 over the interval 0 ≤ 𝜃 ≤ 𝜋 . Express 𝑠 as a definite integral.

  • A 𝑠 = √ 2 𝜃 + 4 𝜃 𝜃 d
  • B 𝑠 = √ 2 𝜃 + 4 𝜃 𝜃 d
  • C 𝑠 = √ 1 + 1 6 𝜃 𝜃 d
  • D 𝑠 = √ 1 + 4 𝜃 𝜃 d
  • E 𝑠 = 2 √ 𝜃 + 4 𝜃 𝜃 d

Using a calculator, or otherwise, find the value of 𝑠 giving your answer to 3 decimal places.

Q19:

Let 𝑠 be the arc length of the polar curve 𝑟 = 𝜃 2 s i n over the interval 0 ≤ 𝜃 ≤ 𝜋 . Express 𝑠 as a definite integral.

  • A 𝑠 = 1 + 𝜃 2 𝜃 2 𝜃 s i n c o s d
  • B 𝑠 = 𝜃 2 1 + 3 𝜃 2 𝜃 s i n c o s d
  • C 𝑠 = 1 + 𝜃 2 𝜃 2 𝜃 s i n c o s d
  • D 𝑠 = 𝜃 2 + 𝜃 2 𝜃 2 𝜃 s i n s i n c o s d
  • E 𝑠 = 𝜃 2 𝜃 s i n d

Using a calculator, or otherwise, find the value of 𝑠 giving your answer to 4 decimal places.

Q20:

Let 𝑠 be the arc length of the polar curve 𝑟 = 2 𝜃 over the interval 𝜋 ≤ 𝜃 ≤ 2 𝜋 . Express 𝑠 as a definite integral.

  • A 𝑠 = 1 + 4 𝜃 𝜃 d
  • B 𝑠 = 2 𝜃 + 2 𝜃 𝜃 d
  • C 𝑠 = 1 + 2 𝜃 𝜃 d
  • D 𝑠 = 2 𝜃 + 1 𝜃 𝜃 d
  • E 𝑠 = 2 𝜃 + 2 𝜃 𝜃 d

Using a calculator, or otherwise, find the value of 𝑠 giving your answer to 4 decimal places.

Q21:

Find a definite integral that represents the arc length of 𝑟 = 4 𝜃 c o s in the interval 0 ≤ 𝜃 ≤ 𝜋 2 .

  • A 4 𝜃 d
  • B 2 √ 𝜃 + 𝜃 𝜃 c o s s i n d
  • C 4 √ 2 𝜃 𝜃 c o s d
  • D 4 𝜃 d
  • E 1 + 1 6 𝜃 𝜃 s i n d

Q22:

Find the arc length of the polar curve 𝑟 = 6 𝜃 + 8 𝜃 s i n c o s , where 𝜃 lies in the interval [ 0 , 𝜋 ] .

  • A 2 0 𝜋
  • B 1 0 0 𝜋
  • C 1 0 𝜋
  • D 5 𝜋
  • E 2 0 0 𝜋

Q23:

Find the arc length of the polar curve 𝑟 = 𝜃 , where 𝜃 lies in the interval [ 0 , 2 𝜋 ] .

  • A 1 6 3 𝜋 + 1 − 1
  • B 4 3 𝜋 + 1 − 1
  • C 1 3 4 𝜋 + 1 − 1
  • D 8 3 𝜋 + 1 − 1
  • E 9 6 𝜋 + 1 6 0 𝜋 1 5

Q24:

Find a definite integral that represents the arc length of 𝑟 = 𝑒 on the interval 0 ≤ 𝜃 ≤ 1 .

  • A √ 2 𝑒 𝜃 d
  • B √ 1 − 𝑒 𝜃 d
  • C √ 1 + 𝑒 𝜃 d
  • D √ 2 𝑒 𝜃 d
  • E √ 2 𝑒 𝜃 d

Q25:

Consider the polar curve 𝑟 = 𝜃 , where 𝜃 lies in the interval [ 0 , 2 𝜋 ] . Find a definite integral that represents the arc length of this curve.

  • A √ 𝜃 + 1 𝜃 d
  • B √ 𝜃 + 1 𝜃 d
  • C 𝜃 + 1 𝜃 d
  • D ( 𝜃 + 1 ) 𝜃 d
  • E √ 𝜃 − 1 𝜃 d

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