Worksheet: Green's Theorem

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In this worksheet, we will practice applying Green’s theorem to evaluate a line integral around a closed curve as the double integral over the plane region bounded by the curve.

Q1:

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

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

Q2:

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

  • A βˆ’ 8 1 5
  • B βˆ’ 1 6 1 5
  • C0
  • D 1 6 1 5
  • E 8 1 5

Q3:

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

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

Q4:

Let 𝐢 be the boundary of the triangle with vertices (0,0),(4,0) and (0,4). Use Green’s Theorem to evaluate 𝑒+𝑦π‘₯+𝑒+π‘₯ο†π‘¦οŒ’ο—οŠ¨ο˜οŠ¨οŽ‘οŽ‘dd, where 𝐢 is traversed counterclockwise.

Q5:

Evaluate ο…‡ο€Ήπ‘₯+𝑦π‘₯+2π‘₯π‘¦π‘¦οŒ’οŠ¨οŠ¨dd, where 𝐢π‘₯=𝑑,𝑦=𝑑:cossin and 0≀𝑑≀2πœ‹.

Q6:

Evaluate ο…‡π‘₯π‘₯+π‘¦π‘¦οŒ’dd, where 𝐢π‘₯=2𝑑,𝑦=3𝑑:cossin and 0≀𝑑≀2π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 𝑒𝑦π‘₯+𝑦+π‘’π‘¦ο…π‘¦οŒ’ο—οŠ©ο—sindcosd, where 𝐢 is traversed counterclockwise.

Q8:

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

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

Q9:

Use Green’s theorem to find ο„Έβ‹…οŒ’Frd, where 𝐢 is the circle of radius π‘Ÿ and center at the origin and F(π‘₯,𝑦)=⟨2π‘₯+5𝑦,2π‘₯+7π‘¦βŸ©.

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

Q10:

The figure shows the graph of 𝑓(π‘₯)=βˆ’3ο€Όπ‘₯+13(π‘₯βˆ’1) over the interval [0,1]. Let 𝑅 be the shaded region and 𝐢 its boundary, traced counterclockwise. Let Fij(π‘₯,𝑦)=𝑦+𝑦.

Use Green’s theorem to calculate ο…‡β‹…οŒ’Frd.

Calculate ο„Έβ‹…οŒ’οŽ Frd, where 𝐢 is the line from π‘Ž to 𝑏.

Calculate ο„Έβ‹…οŒ’οŽ‘Frd, where 𝐢 is the curve from 𝑏 to 𝑐.

Calculate ο„Έβ‹…οŒ’οŽ’Frd, where 𝐢 is the line from 𝑐 to π‘Ž.

Q11:

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 4 5
  • C 1 2
  • D 2 3
  • E 3 4

Consider the vector field F(π‘₯,𝑦)=βŸ¨π‘¦,2π‘₯⟩. What is the function πœ•πœ•π‘₯βˆ’πœ•πœ•π‘¦FF?

Use Green’s theorem to evaluate the line integral ο„Έβ‹…οŒ’Frd, where 𝐢 is the curve above.

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

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