In-class work for 17 July 2012
1. Evaluate the scalar line integral
    a.) $\displaystyle\int_C xy dx + x^2 dy$, where $C$ is the rectangle with vertices $(0,0), (3,0), (3,1),$ and $(0,1)$.
    b.) $\displaystyle\int_C x dx + y dy$, $C$ consists of the line segments from $(0,1)$ to $(0,0)$ and from $(0,0)$ to $(1,0)$ and the parabola $y=1-x^2$ from $(1,0)$ to $(0,1)$
    c.) $\displaystyle\int_C xy^2 dx +2x^2y dy$; $C$ is the triangle with vertices $(0,0), (2,2),$ and $(2,4)$.
    d.) $\displaystyle\int_C xe^{-2x} dx + (x^2+2x^2y^2) dy$; $C$ is the boundary of the region between the circles $x^2+y^2=1$ and $x^2+y^2=4$.

2. Evaluate $\displaystyle\int_C \vec{F} \cdot d\vec{r}$
    a.) $\vec{F}(x,y)=< y^2 \cos x, x^2+2y \sin x >$; $C$ is the triangle from $(0,0)$ to $(2,6)$ to $(2,0)$
    b.) $\vec{F}(x,y) = < x \sin y, y >$; $C$ given by $y=x^2$ form $(-1,1)$ to $(2,4)$.     3.) $\vec{F}(x,y) = < y - \log(x^2+y^2), 2 \arctan(y/x)$; $C$ is the circle $(x-2)^2+(y-3)^2=1$ oriented clockwise


3. A particle starts at the point $(-2.0)$, moves along the $x$-axis to $(2,0)$, and then along the semicircle $y=\sqrt{4-x^2}$ to the starting point Use Green's theorem to find the work done on this particle by the force field $\vec{F}(x,y)=< x,x^3+3xy^2 >$.

4. Evaluate $\displaystyle\int_C xyz ds$ where $C$ given by $x=2 \sin t$, $y=t$, and $ = -2 \cos t$.

5. Find the work done by $\vec{f}=< y^2 \cos z, 2xy \cos z, -xy^2 \sin z >$ in moving a particle over $C$: $\vec{r}(t) = < t^2, \sin t, t >$ for $0 \leq t \leq \pi$.