I have a question that I would like to know the solution
Find the work of strength
$$\vec{F}(x,y) = y\vec{i} + y^2\vec{j}$$
When moving a unit of mass from the point $(0,0)$ to the point $(3,1)$ along the rectilinear segment between these points.
Why is it not possible to apply the formula $\int \vec{F}d\vec{r}$?
Thanks in advance!
Yes we can :
The line between the points $ (0,0) $ and $ (3,1) $ can be parametrised as $$x=0+(3-0)t=3t$$ $$y=0+(1-0)t=t$$
then $$\vec{F}•\vec{dl}=(y,y^2)•(dx,dy)$$ $$=ydx+y^2dy$$ $$t(3dt)+(t^2)(dt)=(3t+t^2)dt$$
thus the work will be
$$W=\int_0^1(3t+t^2)dt=\Bigl[3\frac{t^2}{2}+\frac{t^3}{3}\Bigr]_0^1$$ $$=\frac{11}{6}$$