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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.
    The curve is a triangle with vertices (0,0), (6,3) and (-3,3).  It is traced out with a counter clockwise rotation.
  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.
    There are two circles, both centered at the origin, in the sketch.  The first has a radius of 1 and a counter clockwise rotation.  The second has a radius of 3 and a clockwise rotation.
  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.
    There are four parts to this curve.  The first part is the portion of $y=x^{2]-3$ starting at (-1,-2) and ending at (1,-2).  This is followed by a line starting at (1,-2) and ending at (1,1).  The next part is the portion of $y=2-x^{2}$ starting at (1,1) and ending at (-1,1).  The final part is a line starting at (-1,1) and ending at (-1,-2).
  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.
    The curve is a diamond shaped curve with vertices (-1,-1), (1,1), (1,4) and (-1,2).  It is traced out with a counter clockwise rotation.
  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.
    The curve is a triangle with vertices (0,0), (5,10) and (5,0).  It is traced out with a clockwise rotation.
  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.
    This curve start with the portion of $x^{2}+y^{2}=16$ starting at (0,-4) and ending at (4,0).  This is followed by a line starting at (4,0) and ending at the origin.  The final portion of the curve is a line starting at the origin and ending at (0,-4).
  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.
    This curve start with the portion of $x^{2}+y^{2}=1$ starting at (1,0) and ending at (-1,0).  This is followed by a line starting at (-1,0) and ending at (1,0).
  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.
    The curve is a triangle with vertices (0,0), (1,3) and (1,5).  It is traced out with a clockwise rotation.