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Green's theorem practice problems

Trade a line integral around a closed curve for a double integral inside it. 45 problems with worked solutions; in 45 of them every equation is proved by a computer algebra system.

5 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 12.16

easy
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
  1. Green's theorem: ∮ P dx + Q dy = ∬ (∂Q/∂x − ∂P/∂y) dA.
    Reviewed
  2. \[ \frac{\partial}{\partial x} \left(x + y^{2}\right) - \frac{\partial}{\partial y} \left(x^{2} - y\right) = 2 \]
    ∂Q/∂x − ∂P/∂y.✓ Proved
  3. \[ \int\limits_{0}^{1}\int\limits_{0}^{3} 2\, dx\, dy = 6 \]
    Integrate over the rectangle.✓ Proved
Answer \( 6 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 12.17

easy
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
  1. Green's theorem: ∮ P dx + Q dy = ∬ (∂Q/∂x − ∂P/∂y) dA.
    Reviewed
  2. \[ - \frac{d}{d y} \left(- y^{2}\right) + \frac{\partial}{\partial x} x y = 3 y \]
    ∂Q/∂x − ∂P/∂y.✓ Proved
  3. \[ \int\limits_{0}^{3}\int\limits_{0}^{1} 3 y\, dx\, dy = \frac{27}{2} \]
    Integrate over the rectangle.✓ Proved
Answer \( \frac{27}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 12.19

easy
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
  1. Green's theorem: ∮ P dx + Q dy = ∬ (∂Q/∂x − ∂P/∂y) dA.
    Reviewed
  2. \[ - \frac{\partial}{\partial y} \left(- x y\right) + \frac{\partial}{\partial x} x y = x + y \]
    ∂Q/∂x − ∂P/∂y.✓ Proved
  3. \[ \int\limits_{0}^{2}\int\limits_{0}^{3} \left(x + y\right)\, dx\, dy = 15 \]
    Integrate over the rectangle.✓ Proved
Answer \( 15 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 12.20

easy
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 2] \), counterclockwise.
  1. Green's theorem: ∮ P dx + Q dy = ∬ (∂Q/∂x − ∂P/∂y) dA.
    Reviewed
  2. \[ \frac{d}{d x} 3 x - \frac{d}{d y} y = 2 \]
    ∂Q/∂x − ∂P/∂y.✓ Proved
  3. \[ \int\limits_{0}^{2}\int\limits_{0}^{1} 2\, dx\, dy = 4 \]
    Integrate over the rectangle.✓ Proved
Answer \( 4 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 12.27

easy
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
  1. Green's theorem: ∮ P dx + Q dy = ∬ (∂Q/∂x − ∂P/∂y) dA.
    Reviewed
  2. \[ \frac{d}{d x} x^{2} - \frac{\partial}{\partial y} x y = x \]
    ∂Q/∂x − ∂P/∂y.✓ Proved
  3. \[ \int\limits_{0}^{2}\int\limits_{0}^{3} x\, dx\, dy = 9 \]
    Integrate over the rectangle.✓ Proved
Answer \( 9 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More green's theorem practice

Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
Problem 12.18easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), counterclockwise.
Problem 12.21easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Problem 12.22easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), counterclockwise.
Problem 12.23easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Problem 12.24easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.
Problem 12.25easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
Problem 12.26easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Problem 12.28easy✓ Every equation proved
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Problem 12.29easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
Problem 12.30easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
Problem 12.52easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Problem 12.53easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Problem 12.54easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
Problem 12.55easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Problem 12.56easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
Problem 12.57easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Problem 12.58easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), counterclockwise.
Problem 12.59easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
Problem 12.60easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
Problem 12.61easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 3] \), counterclockwise.
Problem 12.62easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
Problem 12.63easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.
Problem 12.64easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 2] \), counterclockwise.
Problem 12.65easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.
Problem 12.66easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
Problem 12.90easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Problem 12.91easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), counterclockwise.
Problem 12.92easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Problem 12.93easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
Problem 12.94easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 3] \), counterclockwise.
Problem 12.95easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), counterclockwise.
Problem 12.96easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 2] \), counterclockwise.
Problem 12.97easy✓ Every equation proved
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 3] \), counterclockwise.
Problem 12.114easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.
Problem 12.115easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 2] \), counterclockwise.
Problem 12.116easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.
Problem 12.117easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 3] \), counterclockwise.
Problem 12.118easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 3] \), counterclockwise.
Problem 12.119easy✓ Nihil obstat
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 2] \), counterclockwise.
Problem 12.129easy✓ Nihil obstat