Chapter 16.3, Problem 29E

(a)

To determine

Find the work done by the vector field $F=\left(x^2+y\right)i+\left(y^2+x\right)j+z{e}^{z}k$ over the line segment $x=1,y=0,0\le z\le 1$ from $\left(1,0,0\right)$ to $\left(1,0,1\right)$.

(b)

To determine

Find the work done by the vector field $F=\left(x^2+y\right)i+\left(y^2+x\right)j+z{e}^{z}k$ over the helix $r\left(t\right)=\left(\cos t\right)i+\left(\sin t\right)j+\left(\frac{t}{2\pi }\right)k,0\le t\le 2\pi$ from $\left(1,0,0\right)$ to $\left(1,0,1\right)$.

(c)

To determine

Find the work done by the vector field $F=\left(x^2+y\right)i+\left(y^2+x\right)j+z{e}^{z}k$ over the $x\text{-axis}$ from $\left(1,0,0\right)$ to $\left(1,0,1\right)$ followed by the parabola $z=x^2,y=0$ from $\left(0,0,0\right)$ to $\left(1,0,1\right)$.