Chapter 16.2, Problem 7E

(a)

To determine

Determine the line integral of $\text{F}=3y\text{i}+2x\text{j}+4z\text{k}$ from $\left(0,0,0\right)$ to $\left(1,1,1\right)$ over the straight-line path $C_1:\text{r}\left(t\right)=t\text{i}+t\text{j}+t\text{k},0\le t\le 1$.

(b)

To determine

Find the line integral of $\text{F}=3y\text{i}+2x\text{j}+4z\text{k}$ from $\left(0,0,0\right)$ to $\left(1,1,1\right)$ over the curved path $C_2:\text{r}\left(t\right)=t\text{i}+t^2\text{j}+t^4\text{k},0\le t\le 1$.

(c)

To determine

Determine the line integral of $\text{F}=3y\text{i}+2x\text{j}+4z\text{k}$ from $\left(0,0,0\right)$ to $\left(1,1,1\right)$ over the path $C_3\cup C_4$ consisting of the line segment from $\left(0,0,0\right)$ to $\left(1,1,0\right)$ followed by the segment from $\left(1,1,0\right)$ to $\left(1,1,1\right)$.