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Section 16.7 : Green's Theorem

1. Use Green’s Theorem to evaluate \( \displaystyle \int\limits_{C}{{y{x^2}\,dx - {x^2}\,dy}}\) where \(C\) is shown below.
   
   Solution
2. Use Green’s Theorem to evaluate \( \displaystyle \int\limits_{C}{{\left( {6y - 9x} \right)dy - \left( {yx - {x^3}} \right)\,dx}}\) where \(C\) is shown below.
   
   Solution
3. Use Green’s Theorem to evaluate \( \displaystyle \int\limits_{C}{{{x^2}{y^2}\,dx + \left( {y{x^3} + {y^2}} \right)\,dy}}\) where \(C\) is shown below.
   
   Solution
4. Use Green’s Theorem to evaluate \( \displaystyle \int\limits_{C}{{\left( {{y^4} - 2y} \right)\,dx - \left( {6x - 4x{y^3}} \right)\,dy}}\) where \(C\) is shown below.
   
   Solution
5. Verify Green’s Theorem for \( \displaystyle \oint_{C}{{\left( {x{y^2} + {x^2}} \right)\,dx + \left( {4x - 1} \right)\,dy}}\) where \(C\) is shown below by (a)computing the line integral directly and (b) using Green’s Theorem to compute the line integral.
   
   Solution