Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Lesson: Line Integrals in Space

Worksheet • 7 Questions

Q1:

Calculate ο„Έ 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) 𝑠 𝐢 d for the function 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = 𝑧 2 and the curve 𝐢 π‘₯ = 𝑑 𝑑 : s i n , 𝑦 = 𝑑 𝑑 c o s , 𝑧 = 2 √ 2 3 𝑑 3 2 , 0 ≀ 𝑑 ≀ 1 .

  • A 2 5
  • B0
  • C 9 2 0
  • D βˆ’ 2 5
  • E 6 5

Q2:

Calculate ο„Έ 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) 𝑠 𝐢 d for the function 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ 𝑦 + 𝑦 + 2 𝑦 𝑧 and the curve 𝐢 π‘₯ = 𝑑 : 2 , 𝑦 = 𝑑 , 𝑧 = 1 , 1 ≀ 𝑑 ≀ 2 .

  • A 1 3 ο€» 1 7 √ 1 7 βˆ’ 5 √ 5 
  • B 1 3 ο€» 1 7 √ 1 7 + 5 √ 5 
  • C 1 4 ο€» 1 7 √ 1 7 βˆ’ 5 √ 5 
  • D14
  • E 5 6 3

Q3:

Calculate ο„Έ 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) 𝑠 𝐢 d for the function 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = 𝑧 and the curve 𝐢 π‘₯ = 𝑑 : c o s , 𝑦 = 𝑑 s i n , 𝑧 = 𝑑 , 0 ≀ 𝑑 ≀ 2 πœ‹ .

  • A 2 √ 2 πœ‹ 2
  • B 2 √ 2 πœ‹
  • C 2 πœ‹ 2
  • D √ 2 πœ‹ 2
  • E √ 2 πœ‹ 2 2

Q4:

Let be the arc of a unit circle in the -plane traversed counterclockwise from to . Determine the exact value of the line integral of the vector field over .

  • A
  • B
  • C
  • D
  • E

Q5:

Calculate for the vector field and the curve , , , .

Q6:

Calculate for the vector field and the curve , , , .

Q7:

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

  • A 2 4 πœ‹
  • B 2 4 πœ‹ βˆ’ 3
  • C 6 πœ‹
  • D 6 πœ‹ βˆ’ 3
  • E 4 ( 6 πœ‹ βˆ’ 3 )
Preview