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 ο„Έο€Ήπ‘₯+𝑦π‘₯+2π‘₯π‘¦π‘¦οŒ’οŠ¨οŠ¨dd, where 𝐢π‘₯=𝑑,𝑦=2𝑑:, 0≀𝑑≀1, and 𝑑=𝑒sin for 0β‰€π‘’β‰€πœ‹2.

  • A93
  • B133
  • C9
  • D13
  • E26

Q2:

Evaluate ο„Έο€Ήπ‘₯+𝑦π‘₯+2π‘₯π‘¦π‘¦οŒ’οŠ¨οŠ¨dd, where 𝐢π‘₯=𝑑,𝑦=𝑑:cossin and 0β‰€π‘‘β‰€πœ‹.

  • A2πœ‹
  • B0
  • C23
  • Dβˆ’2πœ‹
  • Eβˆ’23

Q3:

Evaluate ο„Έο€Ήπ‘₯+𝑦π‘₯+2π‘₯π‘¦π‘¦οŒ’οŠ¨οŠ¨dd, where 𝐢π‘₯=𝑑,𝑦=2𝑑: and 0≀𝑑≀1.

  • A82
  • B3215
  • C21
  • D9
  • E133

Q4:

Evaluate ο„Έο€Ήπ‘₯+𝑦π‘₯+2π‘₯π‘¦π‘¦οŒ’οŠ¨οŠ¨dd, where 𝐢π‘₯=𝑑,𝑦=2𝑑: and 0≀𝑑≀1.

  • A133
  • B13
  • C93
  • D9
  • E26

Q5:

Evaluate ο„Έο€Ήπ‘₯+𝑦π‘₯+2π‘₯π‘¦π‘¦οŒ’οŠ¨οŠ¨dd, where 𝐢 is the polygonal path from (0,0) to (0,2) to (1,2).

  • A133
  • B203
  • C10
  • D2
  • E5

Q6:

Calculate 𝑓(π‘₯,𝑦)π‘ οŒ’d for the function 𝑓(π‘₯,𝑦) and curve 𝐢, where 𝑓(π‘₯,𝑦)=2π‘₯+𝑦 and 𝐢 is the polygonal path from (0,0) to (3,0) to (3,2).

Q7:

Calculate 𝑓(π‘₯,𝑦)π‘ οŒ’d for the function 𝑓(π‘₯,𝑦) and curve 𝐢, where 𝑓(π‘₯,𝑦)=π‘₯+π‘¦οŠ¨ and 𝐢 is the path from (2,0) counterclockwise along the circle π‘₯+𝑦=4 to the point (βˆ’2,0) and then back to (2,0) along the π‘₯-axis.

  • Aπœ‹
  • B4πœ‹
  • C4(πœ‹+1)
  • D8πœ‹
  • Eπœ‹+4

Q8:

Calculate 𝑓(π‘₯,𝑦)π‘ οŒ’d for the function 𝑓(π‘₯,𝑦) and curve 𝐢, where 𝑓(π‘₯,𝑦)=π‘₯π‘₯+1, 𝐢π‘₯=𝑑,𝑦=0:, and 0≀𝑑≀1.

  • Aln(2)
  • Btantan(2)βˆ’(1)
  • Cln(2)2
  • D2
  • Etantan(2)βˆ’(1)2

Q9:

Calculate 𝑓(π‘₯,𝑦)π‘ οŒ’d for the function 𝑓(π‘₯,𝑦) and curve 𝐢, where 𝑓(π‘₯,𝑦)=π‘₯𝑦, 𝐢π‘₯=𝑑:cos, 𝑦=𝑑sin, and 0β‰€π‘‘β‰€πœ‹2.

  • A12
  • Bβˆ’1
  • Cβˆ’12
  • Dβˆ’14
  • E1

Q10:

Calculate 𝑓(π‘₯,𝑦,𝑧)π‘ οŒ’d for the function 𝑓(π‘₯,𝑦,𝑧)=𝑧 and the curve 𝐢π‘₯=𝑑:cos, 𝑦=𝑑sin, 𝑧=𝑑, 0≀𝑑≀2πœ‹.

  • A2√2πœ‹οŠ¨
  • B√2πœ‹2
  • C2√2πœ‹
  • D√2πœ‹οŠ¨
  • E2πœ‹οŠ¨

Q11:

Calculate 𝑓(π‘₯,𝑦,𝑧)π‘ οŒ’d for the function 𝑓(π‘₯,𝑦,𝑧)=π‘₯𝑦+𝑦+2𝑦𝑧 and the curve 𝐢π‘₯=𝑑:, 𝑦=𝑑, 𝑧=1, 1≀𝑑≀2.

  • A14
  • B14ο€»17√17βˆ’5√5
  • C13ο€»17√17+5√5
  • D13ο€»17√17βˆ’5√5
  • E563

Q12:

Use a line integral to find the lateral surface area of the part of the cylinder π‘₯+𝑦=4 below the plane π‘₯+2𝑦+𝑧=6 and above the π‘₯𝑦-plane.

  • A4(6πœ‹βˆ’3)
  • B6πœ‹βˆ’3
  • C6πœ‹
  • D24πœ‹
  • E24πœ‹βˆ’3

Q13:

Calculate 𝑓(π‘₯,𝑦,𝑧)π‘ οŒ’d for the function 𝑓(π‘₯,𝑦,𝑧)=π‘§οŠ¨ and the curve 𝐢π‘₯=𝑑𝑑:sin, 𝑦=𝑑𝑑cos, 𝑧=2√23π‘‘οŽ’οŽ‘, 0≀𝑑≀1.

  • Aβˆ’25
  • B920
  • C65
  • D25
  • E0

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