Find the work done by the force field $\mathbf{F}(x,y)=-y\mathbf{i}+x\mathbf{j}$ in moving an object along the half-circle $C$ traced out by $$\mathbf{r}(t)=\cos t\,\mathbf{i}+\sin t\,\mathbf{j}\qquad 0\leq t\leq \pi$$

1004

Solution On $C$, $x(t)=\cos t$ and $y(t)=\sin t,$ $0\leq t\leq \pi$. Then $$\begin{eqnarray*} \mathbf{F}( x(t),y(t)) & =&-y(t) \mathbf{i}+x(t) \mathbf{j}=-\sin t\,\mathbf{i}+\cos t\,\mathbf{j}\hbox{}\\ d\mathbf{r}(t)& =&( -\sin t\,\mathbf{i}+\cos t\,\mathbf{j}) \,dt \\[4pt] \mathbf{F}\,{\cdot}\, d\mathbf{r}& =&(\sin ^{2}t+\cos ^{2}t)\,dt=dt \end{eqnarray*}$$

The work $W$ done by $\mathbf{F}$ is $$W=\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\int_{0}^{\pi }dt=\pi$$