Lesson Worksheet: Line Integrals of Vector Fields Mathematics

In this worksheet, we will practice finding the line integral of a vector field along a curve with an orientation.


Suppose 𝐶 is the path given by r(𝑡)=(𝑡,𝑡) for 0𝑡1, 𝐶 is the path given by r(𝑡)=(1𝑡,1𝑡) for 0𝑡1, and Fij=𝑥+(𝑦+1)ln. Without calculating the integrals, which of the following is true?

  • A>FrFrdd
  • B=FrFrdd
  • C<FrFrdd


In the figure, the curve 𝐶 from 𝑃 to 𝑄 consists of two quarter-unit circles, one with center (1, 0) and the other with center (3, 0). Calculate the line integral Frd, where Fij=𝑥2𝑦2𝑥2𝑦2sinsincoscos.

  • A21232+21212sincoscossin
  • B2121221212sincoscossin
  • C2123223212sincoscossin
  • D21212+21212sincoscossin
  • E21232+21212sincoscossin


Let 𝑃 be the arc of a unit circle in the 𝑥𝑦-plane traversed counterclockwise from (0,1) to (1,0). Determine the exact value of the line integral of the vector field Fijk(𝑥,𝑦,𝑧)=3𝑥𝑒+2𝑦𝑧𝑒+𝑦𝑒 over 𝑃.

  • A1+𝑒
  • B1+2𝑒
  • C1𝑒
  • D𝑒1
  • E12𝑒


We explore an example where a vector field F=𝐹,𝐹 satisfies 𝜕𝐹𝜕𝑦𝜕𝐹𝜕𝑥=0 but does not come from a potential. On the plane with the origin removed, consider the vector field F(𝑥,𝑦)=𝑦𝑥+𝑦,𝑥𝑥+𝑦.

On the (open) half-plane 𝑥>0, we can define the angle function 𝜃(𝑥,𝑦)=𝑦𝑥arctan. This is well defined and gives a value between 𝜋2 and 𝜋2. What is the gradient 𝜃?

  • A𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • B𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • C𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • D𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • E𝑥𝑥+𝑦,𝑦𝑥+𝑦

Using the figure shown, use 𝜃 above to define the angle function 𝜃(𝑥,𝑦) on the region 𝑦>0 by a suitable composition with a 𝜋2 rotation.

  • A𝜃(𝑥,𝑦)=𝜃(𝑦,𝑥)+𝜋2
  • B𝜃(𝑥,𝑦)=𝜃(𝑦,𝑥)
  • C𝜃(𝑥,𝑦)=𝜃(𝑦,𝑥)+𝜋2
  • D𝜃(𝑥,𝑦)=𝜃(𝑦,𝑥)+𝜋2
  • E𝜃(𝑥,𝑦)=𝜃(𝑦,𝑥)

What is 𝜃(𝑥,𝑦)?

  • A𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • B𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • C𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • D𝑥𝑥+𝑦,𝑦𝑥+𝑦
  • E𝑦𝑥+𝑦,𝑥𝑥+𝑦

Since 𝜃 and 𝜃 agree on the quadrant 𝑥>0, 𝑦>0, we can define the angle 𝑇(𝑥,𝑦) at each point of the union with values between 𝜋2 and 3𝜋2. Use the functions 𝜃 and 𝜃 to find Frd, where 𝐶 is the arc of the unit circle from 12,32 to 12,12. Answer in terms of 𝜋.

  • A13𝜋12
  • B3𝜋5
  • C𝜋12
  • D𝜋5
  • E12𝜋13

In the same way, we can define 𝜃 on the half-plane 𝑥<0 and 𝜃 on 𝑦<0. Hence, evaluate the line integral Frd around the circle of radius 2, starting from 𝑃(1,1) and going counterclockwise back to 𝑃, stating your answer in terms of 𝜋.

  • A𝜋2
  • B𝜋
  • C𝜋2
  • D2𝜋
  • E2𝜋


Suppose that F is the gradient of the function 𝑓(𝑥,𝑦)=2𝑥𝑦 and we are given points 𝑃(0,0),𝑄(1,0),𝑅(0,1),𝑆(1,1), and 𝑇(1,1). Choose a starting and an end point from this set so as to maximize the integral Frd, where 𝐶 is the line between your chosen points.

  • AFrom 𝑆 to 𝑄
  • BFrom 𝑅 to 𝑇
  • CFrom 𝑇 to 𝑄
  • DFrom 𝑃 to 𝑅
  • EFrom 𝑄 to 𝑇


We will consider two different parameterizations for the line integral Frd, where 𝐿 is the line segment 𝑦=𝑥 from (0,0) to (1,1) and Fij=6𝑥+(𝑥+𝑦).

Calculate the integral using r(𝑡)=(𝑡,𝑡) over 0𝑡1.

  • A6
  • B53
  • C3112
  • D1
  • E236

Calculate the integral using r(𝑡)=𝑒,𝑒 over 0𝑡2ln.

  • A236
  • B12(12)ln
  • C131222ln
  • D1
  • E182343ln

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