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

Homework Help
Start Practicing

Worksheet: Green's Theorem

In this worksheet, we will practice applying Green’s theorem to evaluate line integrals as the area integral of the region it bounds in the plane.

Q1:

Let 𝐢 be the circle with equation π‘₯ + 𝑦 = 1 2 2 . Use Green’s Theorem to evaluate ο…‡ 2 𝑦 π‘₯ βˆ’ 3 π‘₯ 𝑦 𝐢 d d , where 𝐢 is traversed counterclockwise.

  • A 5 πœ‹
  • B βˆ’ 1 0 πœ‹
  • C 1 0 πœ‹
  • D βˆ’ 5 πœ‹
  • E πœ‹

Q2:

Let 𝐢 be the boundary of the triangle with vertices ( 0 , 0 ) , ( 4 , 0 ) and ( 0 , 4 ) . Use Green’s Theorem to evaluate ο…‡ ο€Ί 𝑒 + 𝑦  π‘₯ + ο€Ί 𝑒 + π‘₯  𝑦 𝐢 π‘₯ 2 𝑦 2 2 2 d d , where 𝐢 is traversed counterclockwise.

Q3:

Use Green’s theorem to determine the conditions on π‘Ž , 𝑏 , 𝑐 , and 𝑑 for the vector field F ( π‘₯ , 𝑦 ) = ⟨ π‘Ž π‘₯ + 𝑏 𝑦 , 𝑐 π‘₯ + 𝑑 𝑦 ⟩ to be conservative. In that case, what is the potential function 𝑓 ( π‘₯ , 𝑦 ) for F that satisfies 𝑓 ( 0 , 0 ) = 0 ?

  • A 𝑐 = 𝑏 , 𝑓 ( π‘₯ , 𝑦 ) = π‘Ž 2 π‘₯ + 2 𝑏 π‘₯ 𝑦 + 𝑑 2 𝑦 2 2
  • B 𝑐 = βˆ’ 𝑏 , 𝑓 ( π‘₯ , 𝑦 ) = π‘Ž 2 π‘₯ βˆ’ 𝑏 π‘₯ 𝑦 + 𝑑 2 𝑦 2 2
  • C 𝑐 = βˆ’ 𝑏 , 𝑓 ( π‘₯ , 𝑦 ) = π‘Ž 2 π‘₯ βˆ’ 2 𝑏 π‘₯ 𝑦 + 𝑑 2 𝑦 2 2
  • D 𝑐 = 𝑏 , 𝑓 ( π‘₯ , 𝑦 ) = π‘Ž 2 π‘₯ + 𝑏 π‘₯ 𝑦 + 𝑑 2 𝑦 2 2
  • E 𝑐 = 𝑏 , 𝑓 ( π‘₯ , 𝑦 ) = π‘Ž π‘₯ + 𝑏 π‘₯ 𝑦 + 𝑑 𝑦 2 2

Q4:

Use Green’s theorem to compute the line ο„Έ β‹… 𝐢 F r d , where 𝐢 is the circle of radius π‘Ÿ and center at the origin and F ( π‘₯ , 𝑦 ) = ⟨ 2 π‘₯ + 5 𝑦 , 2 π‘₯ + 7 𝑦 ⟩ .

  • A βˆ’ 5 πœ‹ π‘Ÿ 2
  • B 3 πœ‹ π‘Ÿ 2
  • C 5 πœ‹ π‘Ÿ 2
  • D βˆ’ 3 πœ‹ π‘Ÿ 2
  • E βˆ’ 2 πœ‹ π‘Ÿ 2

Q5:

Use Green’s theorem to evaluate the line integral ο…‡ ο€Ή π‘₯ βˆ’ 𝑦  π‘₯ + 2 π‘₯ 𝑦 𝑦 𝐢 2 2 d d , where 𝐢 is the boundary of the region 𝑅 =  ( π‘₯ , 𝑦 ) 0 ≀ π‘₯ ≀ 1 , 2 π‘₯ ≀ 𝑦 ≀ 2 π‘₯  : 2 , and the curve 𝐢 is traversed counterclockwise.

  • A βˆ’ 1 6 1 5
  • B βˆ’ 8 1 5
  • C 8 1 5
  • D 1 6 1 5
  • E0

Q6:

Evaluate ο…‡ π‘₯ π‘₯ + 𝑦 𝑦 𝐢 d d , where 𝐢 π‘₯ = 2 𝑑 , 𝑦 = 3 𝑑 : c o s s i n and 0 ≀ 𝑑 ≀ 2 πœ‹ .

Q7:

Let 𝐢 be the boundary of the rectangle with vertices ( 1 , βˆ’ 1 ) , ( 1 , 1 ) , ( βˆ’ 1 , 1 ) , and ( βˆ’ 1 , βˆ’ 1 ) . Use Green’s Theorem to evaluate ο…‡ 𝑒 𝑦 π‘₯ + ο€Ή 𝑦 + 𝑒 𝑦  𝑦 𝐢 π‘₯ 3 π‘₯ s i n d c o s d , where 𝐢 is traversed counterclockwise.

Q8:

The figure shows the graph of 𝑓 ( π‘₯ ) = βˆ’ 3 ο€Ό π‘₯ + 1 3  ( π‘₯ βˆ’ 1 ) over the interval [ 0 , 1 ] . Let 𝑅 be the shaded region and 𝐢 its boundary, traced counterclockwise. Let F i j ( π‘₯ , 𝑦 ) = 𝑦 + 𝑦 .

Use Green’s theorem to calculate ο…‡ β‹…  F r d .

Calculate ο„Έ β‹…   F r d , where 𝐢  is the line from π‘Ž to 𝑏 .

Calculate ο„Έ β‹…   F r d , where 𝐢  is the curve from 𝑏 to 𝑐 .

Calculate ο„Έ β‹…   F r d , where 𝐢  is the line from 𝑐 to π‘Ž .

Q9:

The figure shows the steps to producing a curve 𝐢 . It starts as the boundary of the unit square in Figure (a). In Figure (b), we remove a square quarter of the area of the square in (a). In Figure (c), we add a square quarter of the area that we removed in (b). In Figure (d), we remove a square quarter of the area of the square we added in (c). If we continue to do this indefinitely, we will get the curve 𝐢 . We let 𝑅 be the region enclosed by 𝐢 .

By summing a suitable series, find the area of region 𝑅 . Give your answer as a fraction.

  • A 1 4
  • B 3 4
  • C 1 2
  • D 4 5
  • E 2 3

Consider the vector field F ( π‘₯ , 𝑦 ) = ⟨ 𝑦 , 2 π‘₯ ⟩ . What is the function πœ• πœ• π‘₯ βˆ’ πœ• πœ• 𝑦 F F 2 1 ?

Use Green’s theorem to evaluate the line integral ο„Έ β‹… 𝐢 F r d , where 𝐢 is the curve above.

  • A 8 5
  • B 3 4
  • C 4 5
  • D 1 4
  • E 1 5

Q10:

Use Green’s theorem to evaluate the line integral ο…‡ π‘₯ 𝑦 π‘₯ + 2 π‘₯ 𝑦 𝑦 𝐢 2 d d , where 𝐢 is the boundary of 𝑅 =  ( π‘₯ , 𝑦 ) 0 ≀ π‘₯ ≀ 1 , π‘₯ ≀ 𝑦 ≀ π‘₯  : 2 , traversed counterclockwise.

  • A βˆ’ 1 1 2
  • B0
  • C βˆ’ 5 1 2
  • D 1 1 2
  • E 5 1 2

Q11:

Evaluate ο…‡ 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 ≀ 2 πœ‹ .

Related Lessons

Related Videos

Related Worksheets