6.
(a) Let the closed path $C_1$ be the triangle with vertices (1,2), (4,5) and (4,-3) in
the $(x, y)$-plane traversed in the anti-clockwise direction. Sketch the path $C_1$ and
evaluate the integral
$\oint_{C_1} \mathbf{F} \cdot d\mathbf{r}$,
where $\mathbf{F} = (e^{-x} - 3y)\mathbf{i} + (4x - 2y^2)\mathbf{j}$.
(b) Find the area of the region R given in polar coordinates by
$R = \{(r, \theta) \mid 0 \le r \le 2 + \sin(\theta), \ 0 \le \theta \le 2\pi\}$.
(c) Let the curve $C_2$ be the circle in the $(x, y)$-plane given in parametric form by
$x = 2\cos(t)$, $y = 2\sin(t)$ and $t: 0 \to 2\pi$. Evaluate the flux of
$\mathbf{F} = (\sin(2y) + 3x)\mathbf{i} + (4x^3 + 2y)\mathbf{j}$
outwards through $C_2$.
