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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 16.7 : Green's Theorem
- 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.
![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)
- 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.
