Q: Evaluate the closed line intergral $ \oint xdy $ anti-clockwise around the triangle with vetricies $(a,0), (0,0),$ and $(0,b)$
For this section I've reduced the line sections to:
$ C1: x = x, y = 0 ; 0 \leqslant x\leqslant a$
$C2: x=x, y=y; a \leqslant x\leqslant 0 ; 0 \leqslant y\leqslant b$
$C3: x=0, y = y; b \leqslant y\leqslant 0$
However, I'm not sure how to reduce the line sections down to parametrized form to evaluate the integral. Where do go from here? Have I approached this problem incorrectly?
If you have a triangle including the point (0,0) your equations aren't correct, first segment of line from (0,0) to (b,0) i described with equation $y=0$, the second one from point (b,0) to (a,0) is described with line segment $y=-ax/b+a$ and the third line segment is defined with an equation $x=0$, now you can build your integral from thees three parts, with each part having its own parametric equations, only thing you have to take care are the bounds for each line segment, for example line segment 2, is described fully with the following: $$ y=-\frac{a}{b}x+a,x=t,y=-\frac{a}{b}t+a$$ with t in the following bounds: $$ t_1=a,t_2=0 $$ because we are to go anticlockwise around the triangle.