You can use the definition

$\displaystyle \displaystyle{\int_C F\cdot dr}$

Where C is a picewise smooth curve described by the parametric equation $\displaystyle r(t) = x(t) \overset{\rightharpoonup }{i} + y(t) \overset{\rightharpoonup }{j} $ and F is a bidimensional vector field $\displaystyle F(x,y)=P(x,y)\overset{\rightharpoonup }{i} +Q(x,y)\overset{\rightharpoonup }{j} $

By doing $\displaystyle \dfrac{dr}{dt} = r'(t)$ we can rewrite the line integral as:

$\displaystyle \displaystyle{\int_C F(r(t))\cdot r'(t)~dt}$

Where we have a dot product. So for the problem you have $\displaystyle F(x,y) = xy \overset{\rightharpoonup }{i} + x^2 \overset{\rightharpoonup }{j} $ and $\displaystyle r(t)$ will depend, because you should take three line integrals over the three lines that closes the triangle, and each one will have a different parametrization.

So, let $\displaystyle P1=(0,0),~P2=(2,0),~P3=(1,1)$.

We can set the first line $\displaystyle L1$ as the one who links $\displaystyle P1$ to $\displaystyle P2$, $\displaystyle L2$ links $\displaystyle P2$ to $\displaystyle P3$, and $\displaystyle L3$ links $\displaystyle P3$ to $\displaystyle P0$. We followed a counter-clockwise direction.

Now we should parametrize $\displaystyle L1$, $\displaystyle L2$, $\displaystyle L3$.

For $\displaystyle L1$:

You can see that this line has $\displaystyle y=0$ and $\displaystyle 0 \leq x \leq 2$, so maybe the parametrization $\displaystyle r_1(t)=(2t,0)$ with $\displaystyle t\in[0,1]$ should be nice for us.

For $\displaystyle L2$:

This line begins at the $\displaystyle P2$ then go to $\displaystyle P3$, so we want that for some $\displaystyle r(t)$, when t=a we will be at $\displaystyle P2$and when t=b we will be at $\displaystyle P3$. You can use the formula $\displaystyle r(t)=(1-t)P2 + tP3$. It follows that $\displaystyle r_2(t)=(2-t,t)$ with $\displaystyle t\in[0,1]$.

For $\displaystyle L3$:

We can use the same formula again, so $\displaystyle r(t)=(1-t)P3 + tP0$ give us $\displaystyle r_3(t)=(1-t,1-t)$ with $\displaystyle t\in[0,1]$

Now that we have the three parametric equations for each line we can evaluate:

$\displaystyle \displaystyle{\int_{Li} F(r_i(t))\cdot r_i'(t)~dt}$ for $\displaystyle i = 1,2,3$

Finally, we sum each value to get the final answer.

I apologize if I did any mistake, I don't have any book around here by now and I had to remember the whole process - that's why it lacks some deep explanations. I did my best and I hope it helps. Maybe the other guys around here can add some thoughts about it.