Worksheet: Line Integrals in the Plane

In this worksheet, we will practice solving the line integral of a 2-variable function along a parameterized curve in the plane.

Q1:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 𝐶 2 2 d d , where 𝐶 𝑥 = 𝑡 , 𝑦 = 2 𝑡 : , 0 𝑡 1 , and 𝑡 = 𝑢 s i n for 0 𝑢 𝜋 2 .

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

Q2:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 𝐶 2 2 d d , where 𝐶 𝑥 = 𝑡 , 𝑦 = 𝑡 : c o s s i n and 0 𝑡 𝜋 .

  • A0
  • B 2 3
  • C 2 𝜋
  • D 2 3
  • E 2 𝜋

Q3:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 𝐶 2 2 d d , where 𝐶 𝑥 = 𝑡 , 𝑦 = 2 𝑡 : 2 and 0 𝑡 1 .

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

Q4:

Evaluate 𝑥 + 𝑦 d 𝑥 + 2 𝑥 𝑦 d 𝑦 , where 𝐶 : 𝑥 = 𝑡 , 𝑦 = 2 𝑡 and 0 𝑡 1 .

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

Q5:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 𝐶 2 2 d d , where 𝐶 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 𝑓 ( 𝑥 , 𝑦 ) 𝑠 𝐶 d for the function 𝑓 ( 𝑥 , 𝑦 ) and curve 𝐶 , where 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 + 𝑦 and 𝐶 is the polygonal path from ( 0 , 0 ) to ( 3 , 0 ) to ( 3 , 2 ) .

Q7:

Calculate 𝑓 ( 𝑥 , 𝑦 ) 𝑠 𝐶 d for the function 𝑓 ( 𝑥 , 𝑦 ) and curve 𝐶 , where 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 2 and 𝐶 is the path from ( 2 , 0 ) counterclockwise along the circle 𝑥 + 𝑦 = 4 2 2 to the point ( 2 , 0 ) and then back to ( 2 , 0 ) along the 𝑥 -axis.

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

Q8:

Calculate 𝑓 ( 𝑥 , 𝑦 ) 𝑠 d for the function 𝑓 ( 𝑥 , 𝑦 ) and curve 𝐶 , where 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑥 + 1 , 𝐶 𝑥 = 𝑡 , 𝑦 = 0 : , and 0 𝑡 1 .

  • A t a n t a n ( 2 ) ( 1 ) 2
  • B l n ( 2 )
  • C t a n t a n ( 2 ) ( 1 )
  • D l n ( 2 ) 2
  • E2

Q9:

Suppose that F is the gradient of the function 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 𝑦 , and we are given points 𝑃 ( 0 , 0 ) , 𝑄 ( 1 , 0 ) , 𝑅 ( 0 , 1 ) , 𝑆 ( 1 , 1 ) , and 𝑇 ( 1 , 1 ) . Choose a starting and an end point from this set so as to maximize the integral F r d , where 𝐶 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 𝐶 from 𝑃 to 𝑄 consists of two quarter-unit circles, one with centre (1, 0) and the other with centre (3, 0). Calculate the line integral 𝐶 F r d , where F = 𝑥 2 𝑦 2 s i n s i n i j 𝑥 2 𝑦 2 c o s c o s .

  • A 2 1 2 3 2 + 2 1 2 1 2 s i n c o s c o s s i n
  • B 2 1 2 1 2 2 1 2 1 2 s i n c o s c o s s i n
  • C 2 1 2 1 2 + 2 1 2 1 2 s i n c o s c o s s i n
  • D 2 1 2 3 2 + 2 1 2 1 2 s i n c o s c o s s i n
  • E 2 1 2 3 2 2 3 2 1 2 s i n c o s c o s s i n

Q11:

Calculate 𝑓 ( 𝑥 , 𝑦 ) 𝑠 𝐶 d for the function 𝑓 ( 𝑥 , 𝑦 ) and curve 𝐶 , where 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 , 𝐶 𝑥 = 𝑡 : c o s , 𝑦 = 𝑡 s i n , and 0 𝑡 𝜋 2 .

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

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