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Worksheet: Line Integrals of Vector Fields

In this worksheet, we will practice finding the line integral of a vector field along an oriented piecewise differentiable curve in the plane.

Q1:

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

  • A 2 πœ‹
  • B0
  • C πœ‹ 2
  • D βˆ’ 2 πœ‹
  • E βˆ’ πœ‹ 2

Q2:

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

Q3:

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

Q4:

Calculate ο„Έ β‹… 𝐢 f r d for the vector field f ( π‘₯ , 𝑦 ) and curve 𝐢 , where f i j ( π‘₯ , 𝑦 ) = ο€Ή π‘₯ 𝑦  + ο€Ή π‘₯ 𝑦  2 3 and 𝐢 is the polygonal path from ( 0 , 0 ) to ( 1 , 0 ) to ( 0 , 1 ) to ( 0 , 0 ) .

  • A0
  • B 1 3 0
  • C 7 1 5
  • D βˆ’ 1 3 0
  • E βˆ’ 7 1 5

Q5:

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

Q6:

Suppose 𝐢 1 is the path given by r 1 ( 𝑑 ) = ( 𝑑 , 𝑑 ) for 0 ≀ 𝑑 ≀ 1 , 𝐢 2 is the path given by r 2 ( 𝑑 ) = ( 1 βˆ’ 𝑑 , 1 βˆ’ 𝑑 ) for 0 ≀ 𝑑 ≀ 1 , and F i j = π‘₯ + ( 𝑦 + 1 ) 2 l n . Without calculating the integrals, which of the following is true?

  • A ο„Έ β‹… = ο„Έ β‹… 𝐢 𝐢 1 2 F r F r d d
  • B ο„Έ β‹… < ο„Έ β‹… 𝐢 𝐢 1 2 F r F r d d
  • C ο„Έ β‹… > ο„Έ β‹… 𝐢 𝐢 1 2 F r F r d d

Q7:

Calculate ο„Έ β‹… 𝐢 f r d for the vector field f i j k ( π‘₯ , 𝑦 , 𝑧 ) = 𝑦 βˆ’ π‘₯ + 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‹ ( πœ‹ βˆ’ 1 ) 2
  • B 2 πœ‹ 2
  • C 2 πœ‹ ( πœ‹ + 1 )
  • D 2 πœ‹ ( πœ‹ βˆ’ 1 )
  • E 2 πœ‹ ( πœ‹ + 1 ) 2

Q8:

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

  • A0
  • B βˆ’ 2 πœ‹
  • C 2 3 + 2 πœ‹
  • D 2 πœ‹
  • E βˆ’ ο€Ό 2 3 + 2 πœ‹ 

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