Worksheet: Arc Length of a Polar Curve

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.

  • A1𝑒𝑑𝜃
  • B1+𝑒𝑑𝜃
  • C1+𝜃𝑑𝜃
  • D1𝜃𝑑𝜃

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.

  • A4.9348<𝐿<6.5056
  • B9.8696<𝐿<17.1527
  • C4.9348<𝐿<10.8696
  • D4.9348<𝐿<8.0764
  • E8.0764<𝐿<10.8696

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.

  • A4.9348<𝐿<6.5056
  • B4.9348<𝐿<8.0764
  • C6.5056<𝐿<10.8696
  • D4.9348<𝐿<10.8696
  • E6.5056<𝐿<8.0764

Q3:

Let 𝑠 be the arc length of the polar curve 𝑟=3𝜃 over the interval 0𝜃𝜋2. Express 𝑠 as a definite integral.

  • A𝑠=3𝜃+1𝜃d
  • B𝑠=3𝜃+3𝜃d
  • C𝑠=10𝜃d
  • D𝑠=4𝜃d
  • E𝑠=3𝜃+3𝜃d

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=3𝜃sin.

  • A6
  • B3
  • C3𝜋
  • D9𝜋
  • E6𝜋

Q5:

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

  • A22𝜋
  • B2𝜋
  • C22
  • D4𝜋
  • E2𝜋

Q6:

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

  • A𝜃1𝜃𝜃d
  • B0
  • C𝜃+1𝜃𝜃d
  • D𝜃1𝜃𝜃d
  • E𝜃+1𝜃𝜃d

Q7:

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

  • A1+𝜃+𝜃𝜃sincosd
  • B1+𝜃𝜃𝜃sincosd
  • C(1+𝜃+𝜃)𝜃sincosd
  • D(2+2𝜃)𝜃sind
  • E2+2𝜃𝜃sind

Q8:

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

  • A251+(5)5lnln
  • B1+(5)5(251)lnln
  • C2(251)
  • D21+(5)5(251)lnln
  • E1+(5)(251)ln

Q9:

Find a definite integral that represents the arc length of 𝑟=1+𝜃sin on the interval 0𝜃2𝜋.

  • A2+2𝜃𝜃sind
  • B2+2𝜃𝜃sind
  • C1+𝜃𝜃cosd
  • D2+2𝜃𝜃sind
  • E2𝜃+2𝜃𝜃sinsind

Q10:

Find the arc length of the polar curve given by 𝑟=𝑒 on the interval 0𝜃2.

  • A10𝑒1
  • B103𝑒1
  • C103𝑒1
  • D43𝑒1
  • E10𝑒1

Q11:

Find the arc length of the polar curve 𝑟=1𝜃sin over the interval 0𝜃2𝜋.

Q12:

Find the arc length of the polar curve 𝑟=6, where 𝜃 lies in the interval 0,𝜋2.

  • A3𝜋2
  • B6𝜋
  • C3𝜋
  • D6𝜋2
  • E18𝜋

Q13:

Consider the polar curve 𝑟=4𝜃cos, where 𝜃 lies in the interval 0,𝜋2. Find a definite integral that represents the arc length of this curve.

  • A(4𝜃4𝜃)𝜃cossind
  • B16𝜃d
  • C2𝜃𝜃𝜃cossind
  • D2𝜃+𝜃𝜃cossind
  • E4𝜃d

Q14:

Find the arc length of the polar curve 𝑟=6𝜃cos, where 𝜃 lies in the interval 0,𝜋2.

  • A6𝜋
  • B3𝜋
  • C18𝜋
  • D32𝜋
  • E6

Q15:

Find the arc length of the cardioid 𝑟=2+2𝜃cos.

Q16:

Consider the polar curve 𝑟=2𝜃sec, where 𝜃 lies in the interval 0,𝜋3. Find a definite integral that represents the arc length of this curve.

  • A4𝜃+2𝜃𝜃𝜃secsectand
  • B4𝜃𝜃secd
  • C2𝜃2𝜃𝜃𝜃secsectand
  • D2𝜃𝜃secd
  • E2𝜃+2𝜃𝜃𝜃secsectand

Q17:

Find the arc length of the polar curve 𝑟=8+8𝜃cos over the interval 0𝜃𝜋.

Q18:

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

  • A𝑠=2𝜃+4𝜃𝜃d
  • B𝑠=1+16𝜃𝜃d
  • C𝑠=2𝜃+4𝜃𝜃d
  • D𝑠=1+4𝜃𝜃d
  • E𝑠=2𝜃+4𝜃𝜃d

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=𝜃2sin over the interval 0𝜃𝜋. Express 𝑠 as a definite integral.

  • A𝑠=1+𝜃2𝜃2𝜃sincosd
  • B𝑠=𝜃2+𝜃2𝜃2𝜃sinsincosd
  • C𝑠=𝜃21+3𝜃2𝜃sincosd
  • D𝑠=𝜃2𝜃sind
  • E𝑠=1+𝜃2𝜃2𝜃sincosd

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=2𝜃 over the interval 𝜋𝜃2𝜋. Express 𝑠 as a definite integral.

  • A𝑠=2𝜃+2𝜃𝜃d
  • B𝑠=1+2𝜃𝜃d
  • C𝑠=2𝜃+1𝜃𝜃d
  • D𝑠=1+4𝜃𝜃d
  • E𝑠=2𝜃+2𝜃𝜃d

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=4𝜃cos in the interval 0𝜃𝜋2.

  • A2𝜃+𝜃𝜃cossind
  • B1+16𝜃𝜃sind
  • C4𝜃d
  • D42𝜃𝜃cosd
  • E4𝜃d

Q22:

Find the arc length of the polar curve 𝑟=6𝜃+8𝜃sincos, where 𝜃 lies in the interval [0,𝜋].

  • A20𝜋
  • B100𝜋
  • C200𝜋
  • D5𝜋
  • E10𝜋

Q23:

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

  • A83𝜋+11
  • B163𝜋+11
  • C134𝜋+11
  • D43𝜋+11
  • E96𝜋+160𝜋15

Q24:

Find a definite integral that represents the arc length of 𝑟=𝑒 on the interval 0𝜃1.

  • A1+𝑒𝜃d
  • B2𝑒𝜃d
  • C2𝑒𝜃d
  • D2𝑒𝜃d
  • E1𝑒𝜃d

Q25:

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

  • A(𝜃+1)𝜃d
  • B𝜃+1𝜃d
  • C𝜃+1𝜃d
  • D𝜃+1𝜃d
  • E𝜃1𝜃d

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