Evaluate $\oint x\,dx$ over a particular curve $C$ > $$\oint x\,dx\qquad C:\\{x=0,y=0,y=-x+1 \\} $$ **My attempt:** $$\oint x\,dx=\int_{\uparrow}x\,dx+\int_{\nwarrow}x\,dx+\int_{\rightarrow}x\,dx$$ I don't know what should I do now, all the integral limits are from zero too one?

The usual orientation is counter-clockwise so the path $C$, a triangle, would consist of the line segments from $(0,0)$ to $(1,0)$, then from $(1,0)$ to $(0,1)$ and finally from $(0,1)$ back to $(0,0)$.

If you really want to explicitly calculate the line integral, you should parametrize each part. With the given curves, that can be done very easily:

* $(0,0)$ to $(1,0)$: take $x$ as parameter and $y=0$, then $x$ goes from $0$ to $1$;
* $(1,0)$ to $(0,1)$: take $y$ as parameter and then $x=1-y$ with $y$ from $0$ to $1$;
* $(0,1)$ to $(0,0)$: take $y$ as parameter and $x=0$, then $y$ goes from $1$ to $0$; or take $y=-t$ and then $t:0\to 1$.