Green's Theorem:
1. Evaluate the line integral \( \displaystyle{\oint_C y\,e^x \, dx
+ 4\, e^x\, dy }\), where \( C \) is the positively
oriented rectangle with vertices (0, 0), (2, 0), (2, 3) and (0, 3).
solution
2. Evaluate the line integral \( \displaystyle{\oint_C \vec{F}\cdot
\vec{dr} }\), where \( \vec{F}(x,y)=\langle xy, x^2y^3\rangle \) and
\( C \) is the positively oriented triangle with vertices (0, 0),
(1, 0), and (1, 2).
solution
3. Evaluate the line integral \( \displaystyle{\oint_C xy^2\, dx +
4x^2 y \, dy}\), where \( C \) is the positively oriented triangle
with vertices (0, 0), (2, 2), and (2, 4).
solution
4. Find the work done by the force \( {\bf F} = \langle x^2+xy,
y+xy^2
\rangle\) in moving an object from (0, 0) to (1, 1) along the curve \(y
= x^3\),
then back to the origin along
the line \(y=x\).
solution
5. Evaluate the integral \( \displaystyle{\oint_C -2y^3\, dx + 2x^3 \,
dy }\), where \(C\) is the circle \(x^2+y^2=9\).
solution