Work done by multivariable force

Given $$\vec{F} =(y^2+x^2,−2xy)$$. I want to calculate the work done by this force, that is applied on a particle moving from $$(0,0)$$ to $$(3,5)$$ in a straight line.

I know that $$W=\int_1^2 \vec{F} \cdot d\vec{r}$$. The problem is that I don't really now how to properly express $$d\vec{r}$$ and the integral boundaries.

How do I write this down correctly? Is there some sort of useful parameterization (line integral over continuous vector field)?

• You are correct in that you need to parameterize the line. Hint: a line in with slope $m$ can be parameterized as $(t,mt), t \in \mathbb{R}$. – D.B. Feb 28 at 19:06
• I'm new to line integrals and your comment definitely gave me more insight and a very useful hint. Thank you a lot! – Zachary Feb 28 at 19:11

Hint: With $$y=\frac53x$$ we have
$$W=\int_0^3\left(\left[\frac53x\right]^2+x^2,-2x\left [\frac53x\right]\right)\cdot\left(1,\frac53\right)dx$$