How can I determine $\int xy \;ds$ of a triangle with points $(0,0)$, $(1,0)$ and $(1,1)$ *The integral has the letter $C$, which I am not sure how to input here.
I know it may seem easy, but I am not sure of the working
I know that $ds = \sqrt{dx^2 + dy^2}$ but I do not even know what is $dx$ or $dy$ in this case.
Should I split into three segments? What is the simplest method to solve this?
If you're talking about the line integral around the triangle, break it up into three parts corresponding to the three sides of the triangle. On the side from $(0,0)$ to $(1,0)$, $y=0$ so that integral is $0$. The integral from $(1,0)$ to $(1,1)$ has $ds = dy$ and $x=1$. The integral from $(1,1)$ to $(0,0)$, if you use $x$ as parameter, has $y=x$ and $ds = \sqrt{2} dx$.