Worksheet: Line Integrals

In this worksheet, we will practice finding the line integral of a scalar function over a curve in 2D and 3D.

Q1:

Evaluate 𝑥+𝑦𝑥+2𝑥𝑦𝑦dd, where 𝐶𝑥=𝑡,𝑦=2𝑡:, 0𝑡1, and 𝑡=𝑢sin for 0𝑢𝜋2.

  • A 9 3
  • B 1 3 3
  • C9
  • D13
  • E26

Q2:

Evaluate 𝑥+𝑦𝑥+2𝑥𝑦𝑦dd, where 𝐶𝑥=𝑡,𝑦=𝑡:cossin and 0𝑡𝜋.

  • A 2 𝜋
  • B0
  • C 2 3
  • D 2 𝜋
  • E 2 3

Q3:

Evaluate 𝑥+𝑦𝑥+2𝑥𝑦𝑦dd, where 𝐶𝑥=𝑡,𝑦=2𝑡: and 0𝑡1.

  • A82
  • B 3 2 1 5
  • C21
  • D9
  • E 1 3 3

Q4:

Evaluate 𝑥+𝑦𝑥+2𝑥𝑦𝑦dd, where 𝐶𝑥=𝑡,𝑦=2𝑡: and 0𝑡1.

  • A 1 3 3
  • B13
  • C 9 3
  • D9
  • E26

Q5:

Evaluate 𝑥+𝑦𝑥+2𝑥𝑦𝑦dd, where 𝐶 is the polygonal path from (0,0) to (0,2) to (1,2).

  • A 1 3 3
  • B 2 0 3
  • C10
  • D2
  • E5

Q6:

Calculate 𝑓(𝑥,𝑦)𝑠d for the function 𝑓(𝑥,𝑦) and curve 𝐶, where 𝑓(𝑥,𝑦)=2𝑥+𝑦 and 𝐶 is the polygonal path from (0,0) to (3,0) to (3,2).

Q7:

Calculate 𝑓(𝑥,𝑦)𝑠d for the function 𝑓(𝑥,𝑦) and curve 𝐶, where 𝑓(𝑥,𝑦)=𝑥+𝑦 and 𝐶 is the path from (2,0) counterclockwise along the circle 𝑥+𝑦=4 to the point (2,0) and then back to (2,0) along the 𝑥-axis.

  • A 𝜋
  • B 4 𝜋
  • C 4 ( 𝜋 + 1 )
  • D 8 𝜋
  • E 𝜋 + 4

Q8:

Calculate 𝑓(𝑥,𝑦)𝑠d for the function 𝑓(𝑥,𝑦) and curve 𝐶, where 𝑓(𝑥,𝑦)=𝑥𝑥+1, 𝐶𝑥=𝑡,𝑦=0:, and 0𝑡1.

  • A l n ( 2 )
  • B t a n t a n ( 2 ) ( 1 )
  • C l n ( 2 ) 2
  • D2
  • E t a n t a n ( 2 ) ( 1 ) 2

Q9:

Calculate 𝑓(𝑥,𝑦)𝑠d for the function 𝑓(𝑥,𝑦) and curve 𝐶, where 𝑓(𝑥,𝑦)=𝑥𝑦, 𝐶𝑥=𝑡:cos, 𝑦=𝑡sin, and 0𝑡𝜋2.

  • A 1 2
  • B 1
  • C 1 2
  • D 1 4
  • E1

Q10:

Calculate 𝑓(𝑥,𝑦,𝑧)𝑠d for the function 𝑓(𝑥,𝑦,𝑧)=𝑧 and the curve 𝐶𝑥=𝑡:cos, 𝑦=𝑡sin, 𝑧=𝑡, 0𝑡2𝜋.

  • A 2 2 𝜋
  • B 2 𝜋 2
  • C 2 2 𝜋
  • D 2 𝜋
  • E 2 𝜋

Q11:

Calculate 𝑓(𝑥,𝑦,𝑧)𝑠d for the function 𝑓(𝑥,𝑦,𝑧)=𝑥𝑦+𝑦+2𝑦𝑧 and the curve 𝐶𝑥=𝑡:, 𝑦=𝑡, 𝑧=1, 1𝑡2.

  • A14
  • B 1 4 1 7 1 7 5 5
  • C 1 3 1 7 1 7 + 5 5
  • D 1 3 1 7 1 7 5 5
  • E 5 6 3

Q12:

Use a line integral to find the lateral surface area of the part of the cylinder 𝑥+𝑦=4 below the plane 𝑥+2𝑦+𝑧=6 and above the 𝑥𝑦-plane.

  • A 4 ( 6 𝜋 3 )
  • B 6 𝜋 3
  • C 6 𝜋
  • D 2 4 𝜋
  • E 2 4 𝜋 3

Q13:

Calculate 𝑓(𝑥,𝑦,𝑧)𝑠d for the function 𝑓(𝑥,𝑦,𝑧)=𝑧 and the curve 𝐶𝑥=𝑡𝑡:sin, 𝑦=𝑡𝑡cos, 𝑧=223𝑡, 0𝑡1.

  • A 2 5
  • B 9 2 0
  • C 6 5
  • D 2 5
  • E0

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