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Worksheet: Line Integrals in the Plane

Q1:

Evaluate ο„Έ ο€Ή π‘₯ + 𝑦  π‘₯ + 2 π‘₯ 𝑦 𝑦 𝐢 2 2 d d , where 𝐢 π‘₯ = 𝑑 , 𝑦 = 𝑑 : c o s s i n and 0 ≀ 𝑑 ≀ πœ‹ .

  • A0
  • B 2 3
  • C 2 πœ‹
  • D βˆ’ 2 3
  • E βˆ’ 2 πœ‹

Q2:

Evaluate ο„Έ ο€Ή π‘₯ + 𝑦  π‘₯ + 2 π‘₯ 𝑦 𝑦 𝐢 2 2 d d , where 𝐢 π‘₯ = 𝑑 , 𝑦 = 2 𝑑 : 2 and 0 ≀ 𝑑 ≀ 1 .

  • A82
  • B21
  • C 3 2 1 5
  • D 1 3 3
  • E9

Q3:

Evaluate ο„Έ ο€Ή π‘₯ + 𝑦  π‘₯ + 2 π‘₯ 𝑦 𝑦 𝐢 2 2 d d , where 𝐢 π‘₯ = 𝑑 , 𝑦 = 2 𝑑 : and 0 ≀ 𝑑 ≀ 1 .

  • A26
  • B13
  • C 9 3
  • D 1 3 3
  • E9

Q4:

Evaluate ο„Έ ο€Ή π‘₯ + 𝑦  π‘₯ + 2 π‘₯ 𝑦 𝑦 𝐢 2 2 d d , where 𝐢 π‘₯ = 𝑑 , 𝑦 = 2 𝑑 : , 0 ≀ 𝑑 ≀ 1 , and 𝑑 = 𝑒 s i n for 0 ≀ 𝑒 ≀ πœ‹ 2 .

  • A26
  • B13
  • C 9 3
  • D 1 3 3
  • E9

Q5:

Evaluate ο„Έ ο€Ή π‘₯ + 𝑦  π‘₯ + 2 π‘₯ 𝑦 𝑦 𝐢 2 2 d d , where 𝐢 is the polygonal path from ( 0 , 0 ) to ( 0 , 2 ) to ( 1 , 2 ) .

  • A2
  • B5
  • C 2 0 3
  • D 1 3 3
  • E10

Q6:

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

Q7:

Calculate ο„Έ 𝑓 ( π‘₯ , 𝑦 ) 𝑠 𝐢 d for the function 𝑓 ( π‘₯ , 𝑦 ) and curve 𝐢 , where 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ + 𝑦 2 and 𝐢 is the path from ( 2 , 0 ) counterclockwise along the circle π‘₯ + 𝑦 = 4 2 2 to the point ( βˆ’ 2 , 0 ) and then back to ( 2 , 0 ) along the π‘₯ -axis.

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

Q8:

Calculate ο„Έ 𝑓 ( π‘₯ , 𝑦 ) 𝑠 𝐢 d for the function 𝑓 ( π‘₯ , 𝑦 ) and curve 𝐢 , where 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ π‘₯ + 1 2 , 𝐢 π‘₯ = 𝑑 : , and 0 ≀ 𝑑 ≀ 1 .

  • A t a n t a n βˆ’ 1 βˆ’ 1 ( 2 ) βˆ’ ( 1 ) 2
  • B l n ( 2 )
  • C t a n t a n βˆ’ 1 βˆ’ 1 ( 2 ) βˆ’ ( 1 )
  • D l n ( 2 ) 2
  • E2

Q9:

Suppose that F is the gradient of the function 𝑓 ( π‘₯ , 𝑦 ) = 2 π‘₯ βˆ’ 𝑦 3 2 , and we are given points 𝑃 ( 0 , 0 ) , 𝑄 ( 1 , 0 ) , 𝑅 ( 0 , 1 ) , 𝑆 ( 1 , 1 ) , and 𝑇 ( βˆ’ 1 , βˆ’ 1 ) . Choose a starting and an end point from this set so as to maximize the integral ο„Έ β‹… 𝐢 F r d , where 𝐢 is the line between your chosen points.

  • Afrom 𝑃 to 𝑅
  • Bfrom 𝑄 to 𝑇
  • Cfrom 𝑆 to 𝑄
  • Dfrom 𝑇 to 𝑄
  • Efrom 𝑅 to 𝑇

Q10:

In the figure, the curve 𝐢 from 𝑃 to 𝑄 consists of two quarter-unit circles, one with center (1, 0) and the other with center (3, 0). Calculate the line integral ο„Έ β‹… 𝐢 F r d , where F = ο€» π‘₯ 2  ο€» 𝑦 2  s i n s i n i j βˆ’ ο€» π‘₯ 2  ο€» 𝑦 2  c o s c o s .

  • A 2 ο€Ό 1 2  ο€Ό 3 2  + 2 ο€Ό 1 2  ο€Ό 1 2  s i n c o s c o s s i n
  • B βˆ’ 2 ο€Ό 1 2  ο€Ό 1 2  βˆ’ 2 ο€Ό 1 2  ο€Ό 1 2  s i n c o s c o s s i n
  • C βˆ’ 2 ο€Ό 1 2  ο€Ό 1 2  + 2 ο€Ό 1 2  ο€Ό 1 2  s i n c o s c o s s i n
  • D 2 ο€Ό 1 2  ο€Ό 3 2  + 2 ο€Ό 1 2  ο€Ό 1 2  s i n c o s c o s s i n
  • E βˆ’ 2 ο€Ό 1 2  ο€Ό 3 2  βˆ’ 2 ο€Ό 3 2  ο€Ό 1 2  s i n c o s c o s s i n

Q11:

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

  • A1
  • B βˆ’ 1 2
  • C βˆ’ 1
  • D 1 2
  • E βˆ’ 1 4