(Solved): can someone please explain how to figure out the parametric equations (highlighted) Calculate the ve ...

Calculate the vector line integral $${?}_C\mathbf{F}?d\mathbf{r}$$, where $$\mathbf{F}=?x?y,y?2x?$$ and $$C$$ is the counterclockwise-oriented rectangle with vertices $$(0,0),(2,0),(2,3)$$, and $$(0,3)$$ Well done! You got it right. $$?6$$ Answer Explanation Correct answers: $$?6$$ The curve $$C$$ is the union of its four sides, and each side is smooth. Thus, $$C$$ is piecewise smooth. Let $$C_1$$ represent the side from $$(0,0)$$ to $$(2,0)$$, let $$C_2$$ represent the side from $$(2,0)$$ to $$(2,3)$$, let $$C_3$$ represent the side from $$(2,3)$$ to $$(0,3)$$, and let $$C_4$$ represent the side from $$(0,3)$$ to $$(0,0)$$. Then, $${?}_C\mathbf{F}?d\mathbf{r}={?}_{C_1}\mathbf{F}?d\mathbf{r}+{?}_{C_2}\mathbf{F}?d\mathbf{r}+{?}_{C_3}\mathbf{F}?d\mathbf{r}+{?}_{C_4}\mathbf{F}?d\mathbf{r}$$ Now parameterize each side of the rectangle. Note that the rectangle is being traversed counterclockwise: $$\begin{array}{l}{C}_1:?t,0?,0?t?2\\ C_2:?2,t?,0?t?3\\ C_3:?2?t,3?,0?t?2\\ C_4:?0,3?t?,0?t?3\end{array}$$