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Lesson: Finding the Length of a Polar Curve

Worksheet • 25 Questions

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

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.

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