Example 2
Evaluate the work done $W=\int_{O}^{P} \mathbf{F} \cdot d \mathbf{r}=\int_{O}^{P}\left(F_{x} d x+F_{y} d y\right)$ by the two-dimensional force $\mathbf{F}=\left(x^{2}, 2 x y\right)$ along the three paths joining the origin to the point $P=(1,1)$ as shown in Figure $4.24(\mathrm{a})$ and defined as follows: (a) This path goes along the $x$ axis to $Q=(1,0)$ and then straight up to $P .$ (Divide the integral into two pieces, $\int_{O}^{P}=\int_{O}^{Q}+\int_{Q}^{P}) .$ (b) On this path $y=x^{2},$ and you can replace the term $d y$ in (4.100) by $d y=2 x d x$ and convert the whole integral into an integral over $x$. (c) This path is given parametrically as $x=t^{3}, y=t^{2} .$ In this case rewrite $x, y, d x,$ and $d y$ in (4.100) in terms of $t$ and $d t,$ and convert the integral into an integral over $t$.