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### Section 5-7 : Green's Theorem

1. Use Green’s Theorem to evaluate $$\displaystyle \int\limits_{C}{{\left( {y{x^2} - y} \right)\,dx + \left( {{x^3} + 4} \right)\,dy}}$$ where $$C$$ is shown below. 2. Use Green’s Theorem to evaluate $$\displaystyle \int\limits_{C}{{\left( {7x + {y^2}} \right)dy - \left( {{x^2} - 2y} \right)\,dx}}$$ where $$C$$ is are the two circles as shown below. 3. Use Green’s Theorem to evaluate $$\displaystyle \int\limits_{C}{{\left( {{y^2} - 6y} \right)\,dx + \left( {{y^3} + 10{y^2}} \right)\,dy}}$$ where $$C$$ is shown below. 4. Use Green’s Theorem to evaluate $$\displaystyle \int\limits_{C}{{x{y^2}\,dx + \left( {1 - x{y^3}} \right)\,dy}}$$ where $$C$$ is shown below. 5. Use Green’s Theorem to evaluate $$\displaystyle \oint_{C}{{\left( {{y^2} - 4x} \right)\,dx - \left( {2 + {x^2}{y^2}} \right)\,dy}}$$ where $$C$$ is shown below. 6. Use Green’s Theorem to evaluate $$\displaystyle \oint_{C}{{\left( {{y^3} - x{y^2}} \right)\,dx + \left( {2 - {x^3}} \right)\,dy}}$$ where $$C$$ is shown below. 7. Verify Green’s Theorem for $$\displaystyle \oint_{C}{{\left( {6 + {x^2}} \right)\,dx + \left( {1 - 2xy} \right)\,dy}}$$ where $$C$$ is shown below by (a) computing the line integral directly and (b) using Green’s Theorem to compute the line integral. 8. Verify Green’s Theorem for $$\displaystyle \oint_{C}{{\left( {6y - 3{y^2} + x} \right)\,dx + y{x^3}dy}}$$ where $$C$$ is shown below by (a) computing the line integral directly and (b) using Green’s Theorem to compute the line integral. 