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Worksheet: Finding the Length of a Polar Curve

In this worksheet, we will practice using the polar arc length formula for a parametric curve to find its length.

Q1:

Write the integral for the arc length of the spiral π‘Ÿ = πœƒ between πœƒ = 0 and πœƒ = πœ‹ . Do not evaluate the integral.

  • A ο„Έ √ 1 βˆ’ πœƒ 𝑑 πœƒ πœ‹ 0 2
  • B ο„Έ √ 1 + 𝑒 𝑑 πœƒ πœ‹ 0 πœƒ 2
  • C ο„Έ √ 1 βˆ’ 𝑒 𝑑 πœƒ πœ‹ 0 πœƒ 2
  • D ο„Έ √ 1 + πœƒ 𝑑 πœƒ πœ‹ 0 2

Q2:

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

Use the fact that π‘₯ < 1 + π‘₯ < ( 1 + π‘₯ ) 2 2 2 when π‘₯ > 0 to find lower and upper bounds for the length 𝐿 of the spiral π‘Ÿ = πœƒ between πœƒ = 0 and πœƒ = πœ‹ . Give your answer to 4 decimal places.

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

By comparing √ 1 + π‘₯ 2 to the average of π‘₯ and 1 + π‘₯ when π‘₯ > 0 , find better bounds for estimating 𝐿 . 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 6 . 5 0 5 6 < 𝐿 < 8 . 0 7 6 4
  • D 4 . 9 3 4 8 < 𝐿 < 8 . 0 7 6 4
  • E 6 . 5 0 5 6 < 𝐿 < 1 0 . 8 6 9 6

Q3:

Let be the arc length of the polar curve over the interval . Express as a definite integral.

  • A
  • B
  • C
  • D
  • E

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

Q4:

Find the total arc length of .

  • A
  • B
  • C6
  • D
  • E3

Q5:

Find the arc length of the polar curve π‘Ÿ = πœƒ + πœƒ s i n c o s , where πœƒ lies in the interval [ 0 , πœ‹ ] .

  • A 2 √ 2 πœ‹
  • B 2 √ 2
  • C 2 πœ‹
  • D √ 2 πœ‹
  • E 4 πœ‹

Q6:

Consider the polar curve π‘Ÿ = 1 πœƒ , where πœƒ lies in the interval [ 0 , 2 πœ‹ ] . Find a definite integral that represents the arc length of this curve.

  • A ο„Έ √ πœƒ βˆ’ 1 πœƒ πœƒ 2 πœ‹ 0 d
  • B ο„Έ πœƒ + 1 πœƒ πœƒ 2 πœ‹ 0 2 2 d
  • C0
  • D ο„Έ √ πœƒ + 1 πœƒ πœƒ 2 πœ‹ 0 2 2 d
  • E ο„Έ √ πœƒ βˆ’ 1 πœƒ πœƒ 2 πœ‹ 0 2 2 d

Q7:

Consider the polar curve π‘Ÿ = 1 + πœƒ s i n , where πœƒ lies in the interval [ 0 , 2 πœ‹ ] . Find a definite integral that represents the arc length of this curve.

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

Q8:

Find the arc length of the polar curve π‘Ÿ = 5 πœƒ , where πœƒ lies in the interval [ 0 , 2 πœ‹ ] .

  • A √ 1 + ( 5 ) ( 2 5 βˆ’ 1 ) l n 2 πœ‹
  • B √ 2 ( 2 5 βˆ’ 1 ) πœ‹
  • C 2 5 √ 1 + ( 5 ) 5 πœ‹ 2 l n l n
  • D √ 1 + ( 5 ) 5 ( 2 5 βˆ’ 1 ) l n l n 2 πœ‹
  • E 2 √ 1 + ( 5 ) 5 ( 2 5 βˆ’ 1 ) l n l n 2 πœ‹

Q9:

Find a definite integral that represents the arc length of π‘Ÿ = 1 + πœƒ s i n on the interval 0 ≀ πœƒ ≀ 2 πœ‹ .

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

Q10:

Find the arc length of the polar curve given by π‘Ÿ = 𝑒 3 πœƒ on the interval 0 ≀ πœƒ ≀ 2 .

  • A √ 1 0 3 ο€Ή 𝑒 βˆ’ 1  6 πœ‹
  • B 4 3 ο€Ή 𝑒 βˆ’ 1  3
  • C 1 0 ο€Ή 𝑒 βˆ’ 1  1 2
  • D √ 1 0 3 ο€Ή 𝑒 βˆ’ 1  6
  • E √ 1 0 ο€Ή 𝑒 βˆ’ 1  6

Q11:

Find the arc length of the polar curve π‘Ÿ = 1 βˆ’ πœƒ s i n over the interval 0 ≀ πœƒ ≀ 2 πœ‹ .

Q12:

Find the arc length of the polar curve π‘Ÿ = 6 , where πœƒ lies in the interval  0 , πœ‹ 2  .

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

Q13:

Consider the polar curve π‘Ÿ = 4 πœƒ c o s , where πœƒ lies in the interval  0 , πœ‹ 2  . Find a definite integral that represents the arc length of this curve.

  • A ο„Έ 2 √ πœƒ βˆ’ πœƒ πœƒ πœ‹ 2 0 c o s s i n d
  • B ο„Έ 1 6 πœƒ πœ‹ 2 0 d
  • C ο„Έ ( 4 πœƒ βˆ’ 4 πœƒ ) πœƒ πœ‹ 2 0 c o s s i n d
  • D ο„Έ 4 πœƒ πœ‹ 2 0 d
  • E ο„Έ 2 √ πœƒ + πœƒ πœƒ πœ‹ 2 0 c o s s i n d

Q14:

Find the arc length of the polar curve π‘Ÿ = 6 πœƒ c o s , where πœƒ lies in the interval  0 , πœ‹ 2  .

  • A 3 2 πœ‹
  • B 1 8 πœ‹
  • C6
  • D 3 πœ‹
  • E 6 πœ‹

Q15:

Find the arc length of the cardioid π‘Ÿ = 2 + 2 πœƒ c o s .

Q16:

Consider the polar curve π‘Ÿ = 2 πœƒ s e c , where πœƒ lies in the interval  0 , πœ‹ 3  . Find a definite integral that represents the arc length of this curve.

  • A ο„Έ √ 2 πœƒ + 2 πœƒ πœƒ πœƒ πœ‹ 3 0 s e c s e c t a n d
  • B ο„Έ 4 πœƒ πœƒ πœ‹ 3 0 4 s e c d
  • C ο„Έ √ 4 πœƒ + 2 πœƒ πœƒ πœƒ πœ‹ 3 0 2 s e c s e c t a n d
  • D ο„Έ 2 πœƒ πœƒ πœ‹ 3 0 2 s e c d
  • E ο„Έ √ 2 πœƒ βˆ’ 2 πœƒ πœƒ πœƒ πœ‹ 3 0 s e c s e c t a n d

Q17:

Find the arc length of the polar curve π‘Ÿ = 8 + 8 πœƒ c o s over the interval 0 ≀ πœƒ ≀ πœ‹ .

Q18:

Let 𝑠 be the arc length of the polar curve π‘Ÿ = 2 πœƒ 2 over the interval 0 ≀ πœƒ ≀ πœ‹ . Express 𝑠 as a definite integral.

  • A 𝑠 = ο„Έ √ 2 πœƒ + 4 πœƒ πœƒ πœ‹ 0 2 d
  • B 𝑠 = ο„Έ √ 1 + 4 πœƒ πœƒ πœ‹ 0 d
  • C 𝑠 = ο„Έ √ 2 πœƒ + 4 πœƒ πœƒ πœ‹ 0 4 2 d
  • D 𝑠 = ο„Έ 2 √ πœƒ + 4 πœƒ πœƒ πœ‹ 0 4 2 d
  • E 𝑠 = ο„Έ √ 1 + 1 6 πœƒ πœƒ πœ‹ 0 2 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 π‘Ÿ = ο€½ πœƒ 2  s i n 2 over the interval 0 ≀ πœƒ ≀ πœ‹ . Express 𝑠 as a definite integral.

  • A 𝑠 = ο„Έ ο„Ÿ ο€½ πœƒ 2  + ο€½ πœƒ 2  ο€½ πœƒ 2  πœƒ πœ‹ 0 s i n s i n c o s d
  • B 𝑠 = ο„Έ ο„Ÿ 1 + ο€½ πœƒ 2  ο€½ πœƒ 2  πœƒ πœ‹ 0 s i n c o s d
  • C 𝑠 = ο„Έ ο€½ πœƒ 2  ο„Ÿ 1 + 3 ο€½ πœƒ 2  πœƒ πœ‹ 0 2 s i n c o s d
  • D 𝑠 = ο„Έ ο€½ πœƒ 2  πœƒ πœ‹ 0 s i n d
  • E 𝑠 = ο„Έ ο„Ÿ 1 + ο€½ πœƒ 2  ο€½ πœƒ 2  πœƒ πœ‹ 0 2 2 s i n c o s 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 π‘Ÿ = 2 πœƒ over the interval πœ‹ ≀ πœƒ ≀ 2 πœ‹ . Express 𝑠 as a definite integral.

  • A 𝑠 = ο„Έ ο„ž 2 πœƒ + 2 πœƒ πœƒ 2 πœ‹ πœ‹ 2 d
  • B 𝑠 = ο„Έ ο„ž 1 + 2 πœƒ πœƒ 2 πœ‹ πœ‹ 2 d
  • C 𝑠 = ο„Έ ο„ž 2 πœƒ + 2 πœƒ πœƒ 2 πœ‹ πœ‹ 2 4 d
  • D 𝑠 = ο„Έ 2 ο„ž πœƒ + 1 πœƒ πœƒ 2 πœ‹ πœ‹ 2 4 d
  • E 𝑠 = ο„Έ ο„ž 1 + 4 πœƒ πœƒ 2 πœ‹ πœ‹ 4 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 π‘Ÿ = 4 πœƒ c o s in the interval 0 ≀ πœƒ ≀ πœ‹ 2 .

  • A ο„Έ  1 + 1 6 πœƒ πœƒ πœ‹ 2 0 2 s i n d
  • B ο„Έ 4 πœƒ 2 πœ‹ 0 d
  • C 2 ο„Έ √ πœƒ + πœƒ πœƒ πœ‹ 2 0 c o s s i n d
  • D ο„Έ 4 πœƒ πœ‹ 2 0 d
  • E ο„Έ 4 √ 2 πœƒ πœƒ πœ‹ 2 0 c o s d

Q22:

Find the arc length of the polar curve π‘Ÿ = 6 πœƒ + 8 πœƒ s i n c o s , where πœƒ lies in the interval [ 0 , πœ‹ ] .

  • A 1 0 0 πœ‹
  • B 2 0 πœ‹
  • C 2 0 0 πœ‹
  • D 1 0 πœ‹
  • E 5 πœ‹

Q23:

Find the arc length of the polar curve π‘Ÿ = πœƒ 2 , where πœƒ lies in the interval [ 0 , 2 πœ‹ ] .

  • A 1 6 3  ο€Ή πœ‹ + 1  βˆ’ 1  2 3 2
  • B 1 3  ο€Ή 4 πœ‹ + 1  βˆ’ 1  2 3 2
  • C 4 3  ο€Ή πœ‹ + 1  βˆ’ 1  2 3 2
  • D 8 3  ο€Ή πœ‹ + 1  βˆ’ 1  2 3 2
  • E 9 6 πœ‹ + 1 6 0 πœ‹ 1 5 5 3

Q24:

Find a definite integral that represents the arc length of π‘Ÿ = 𝑒 πœƒ on the interval 0 ≀ πœƒ ≀ 1 .

  • A ο„Έ √ 1 + 𝑒 πœƒ 1 0 2 πœƒ d
  • B ο„Έ √ 2 𝑒 πœƒ 2 πœ‹ 0 πœƒ d
  • C ο„Έ √ 1 βˆ’ 𝑒 πœƒ 1 0 2 πœƒ d
  • D ο„Έ √ 2 𝑒 πœƒ 1 0 πœƒ d
  • E ο„Έ √ 2 𝑒 πœƒ 1 0 πœƒ d

Q25:

Consider the polar curve π‘Ÿ = πœƒ , where πœƒ lies in the interval [ 0 , 2 πœ‹ ] . Find a definite integral that represents the arc length of this curve.

  • A ο„Έ √ πœƒ + 1 πœƒ 2 πœ‹ 0 d
  • B ο„Έ ο€Ή πœƒ + 1  πœƒ 2 πœ‹ 0 2 d
  • C ο„Έ ( πœƒ + 1 ) πœƒ 2 πœ‹ 0 d
  • D ο„Έ √ πœƒ + 1 πœƒ 2 πœ‹ 0 2 d
  • E ο„Έ √ πœƒ βˆ’ 1 πœƒ 2 πœ‹ 0 2 d

Q26:

Find the arc length of the polar curve given by the equation π‘Ÿ = 6 on the interval 0 ≀ πœƒ ≀ πœ‹ 2 .

  • A 1 8 πœ‹
  • B πœ‹ 2
  • C 9 πœ‹
  • D 3 πœ‹
  • E 1 2 πœ‹

Q27:

Find the arc length of the polar curve π‘Ÿ = 𝑒 3 πœƒ , where πœƒ lies in the interval [ 0 , 2 ] .

  • A √ 2 3 ο€Ή 𝑒 βˆ’ 1  6
  • B √ 1 0 𝑒 3 6
  • C √ 1 0 ο€Ή 𝑒 βˆ’ 1  6
  • D √ 1 0 3 ο€Ή 𝑒 βˆ’ 1  6
  • E 2 3 ο€Ή 𝑒 βˆ’ 1  6

Q28:

Consider the polar curve π‘Ÿ = 𝑒 πœƒ , where πœƒ lies in the interval [ 0 , 1 ] . Find a definite integral that represents the arc length of this curve.

  • A ο„Έ √ 2 𝑒 πœƒ 1 0 πœƒ d
  • B ο„Έ 2 𝑒 πœƒ 1 0 2 πœƒ d
  • C ο„Έ 2 𝑒 πœƒ 1 0 πœƒ d
  • D ο„Έ √ 2 𝑒 πœƒ 1 0 πœƒ d
  • E ο„Έ √ 2 𝑒 πœƒ 1 0 πœƒ 2 d

Q29:

Find the length of on the interval .

  • A
  • B
  • C
  • D
  • E

Q30:

Find a definite integral that represents the arc length of π‘Ÿ = 2 πœƒ s e c on the interval 0 ≀ πœƒ ≀ πœ‹ 3 .

  • A ο„Έ √ 1 + 4 πœƒ πœƒ πœ‹ 3 0 2 s e c d
  • B ο„Έ 2 πœƒ πœƒ 2 πœ‹ 0 2 s e c d
  • C ο„Έ 2 πœƒ √ 1 βˆ’ πœƒ πœƒ πœ‹ 3 0 2 s e c t a n d
  • D ο„Έ 2 πœƒ πœƒ πœ‹ 3 0 2 s e c d
  • E ο„Έ  2 πœƒ ο€Ή 1 βˆ’ πœƒ  πœƒ πœ‹ 3 0 2 s e c t a n d

Q31:

Let 𝑠 be the arc length of the polar curve π‘Ÿ = ( 3 πœƒ ) s i n c o s over the interval 0 ≀ πœƒ ≀ πœ‹ . Express 𝑠 as a definite integral.

  • A 𝑠 = ο„Έ √ ( 3 πœƒ ) + 3 πœƒ ( 3 πœƒ ) πœƒ πœ‹ 0 s i n c o s s i n c o s c o s d
  • B 𝑠 = ο„Έ √ 1 + 3 πœƒ ( 3 πœƒ ) πœƒ πœ‹ 0 s i n c o s c o s d
  • C 𝑠 = ο„Έ  ( 3 πœƒ ) + πœƒ ( 3 πœƒ ) πœƒ πœ‹ 0 2 2 2 s i n c o s s i n c o s c o s d
  • D 𝑠 = ο„Έ  ( 3 πœƒ ) + 9 πœƒ ( 3 πœƒ ) πœƒ πœ‹ 0 2 2 2 s i n c o s s i n c o s c o s d
  • E 𝑠 = ο„Έ  1 + πœƒ ( 3 πœƒ ) πœƒ πœ‹ 0 2 2 s i n c o s c o s d

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

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