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.
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.
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.
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.
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.
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.
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.
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.
![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).](GreensTheorem_Files/image003.gif)
