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Worksheet: Line Integrals in Space

Q1:

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

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 5 6 3
  • B 1 4 ο€» 1 7 √ 1 7 βˆ’ 5 √ 5 
  • C14
  • D 1 3 ο€» 1 7 √ 1 7 βˆ’ 5 √ 5 
  • E 1 3 ο€» 1 7 √ 1 7 + 5 √ 5 

Q3:

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

Q4:

Let 𝑃 be the arc of a unit circle in the π‘₯ 𝑦 -plane traversed counterclockwise from ( 0 , 1 ) to ( 1 , 0 ) . Determine the exact value of the line integral of the vector field F i j k ( π‘₯ , 𝑦 , 𝑧 ) = 3 π‘₯ 𝑒 + 2 𝑦 𝑧 𝑒 + 𝑦 𝑒 2 π‘₯ + 𝑦 𝑧 π‘₯ + 𝑦 𝑧 2 π‘₯ + 𝑦 𝑧 3 2 3 2 3 2 over 𝑃 .

  • A 1 βˆ’ 𝑒
  • B 1 βˆ’ 2 𝑒
  • C 1 + 𝑒
  • D 𝑒 βˆ’ 1
  • E 1 + 2 𝑒

Q5:

Calculate ο„Έ β‹… 𝐢 f r d for the vector field f i j k ( π‘₯ , 𝑦 , 𝑧 ) = βˆ’ + and the curve 𝐢 ∢ π‘₯ = 3 𝑑 , 𝑦 = 2 𝑑 , 𝑧 = 𝑑 , 0 ≀ 𝑑 ≀ 1 .

Q6:

Calculate ο„Έ β‹… 𝐢 f r d for the vector field f i j k ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ + 𝑦 + 𝑧 and the curve 𝐢 π‘₯ = 𝑑 : c o s , 𝑦 = 𝑑 s i n , 𝑧 = 2 , 0 ≀ 𝑑 ≀ 2 πœ‹ .

Q7:

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 6 5
  • B 9 2 0
  • C βˆ’ 2 5
  • D 2 5
  • E0