Worksheet: Arc Length of Vector-Valued Functions

In this worksheet, we will practice finding the length of the curve of a vector-valued function and using that to find the arc-length parameterization of a curve.


Calculate the arc length of 𝑓(𝑡)=23𝑡,23𝑡,2𝑡cossin over the given interval [0,1].

  • A292
  • B258
  • C58
  • D23
  • E92


Calculate the arc length of 𝑓(𝑡)=(32𝑡,32𝑡,3𝑡)cossin over the given interval 0,𝜋2.

  • A6𝜋
  • B𝜋52
  • C3𝜋62
  • D3𝜋52
  • E𝜋62


Calculate the arc length of 𝑓(𝑡)=𝑡+1𝑡,𝑡+1𝑡,22𝑡cossin over the given interval [0,1].

  • A2
  • B52
  • C103
  • D4
  • E72


Parametrize the curve 𝑓(𝑡)=(32𝑡,32𝑡,3𝑡)cossin for 𝑡 in 0,𝜋2 using arc length parameterization.

  • A𝑓(𝑠)=35𝑠15,35𝑠15,5𝑠5cossin for all 𝑠 in 0,3𝜋52
  • B𝑓(𝑠)=32𝑠3,32𝑠3,2𝑠3cossin for all 𝑠 in 0,3𝜋2
  • C𝑓(𝑠)=325𝑠15,325𝑠15,5𝑠5cossin for all 𝑠 in 0,3𝜋52
  • D𝑓(𝑠)=325𝑠15,325𝑠15,5𝑠5cossin for all 𝑠 in 0,𝜋2
  • E𝑓(𝑠)=3𝑠3,3𝑠3,2𝑠3cossin for all 𝑠 in 0,3𝜋2


Determine the arc length parameterization of the curve 𝑓(𝑡)=23𝑡,23𝑡,2𝑡cossin for 𝑡 in [0,1].

  • A𝑓(𝑠)=2𝑠+1624,2𝑠+1624,2𝑠+1624cossin for all 𝑠 in 0,259
  • B𝑓(𝑠)=23𝑠212,23𝑠212,𝑠16cossin for all 𝑠 in [0,1]
  • C𝑓(𝑠)=2𝑠+1624,2𝑠+1624,𝑠cossin for all 𝑠 in 0,259
  • D𝑓(𝑠)=2𝑠24,2𝑠24,𝑠16cossin for all 𝑠 in 16,25
  • E𝑓(𝑠)=2𝑠24,2𝑠212,𝑠4cossin for all 𝑠 in [0,1]

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