Question

Find the work required to move an object in the force field F = e^x + y (1, 1, z) along the straight line from A(0, 0, 0) to B(-4, 5, -5). Check to see if the force is conservative. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) A. The force is not conservative. The work is ___. B. The force is conservative. The work is _____ .

Answer 1

$\vec F(x,y,z)=(e^x+y)(1,1,z)$

is conservative if we can find a scalar function $f$ such that $\nabla f=\vec F$. This would require

$f_x=e^x+y$

$f_y=e^x+y$

$f_z=(e^x+y)z$

Integrating both sides of the first equation wrt $x$ gives

$f(x,y,z)=e^x+xy+g(y,z)$

Differentiating both sides of this wrt $y$ gives

$f_y=x+g_y=1\implies g_y=1-x\implies g(y,z)=y-xy+h(z)$

but we assumed $g$ was a function of $y$ and $z$, independent of $x$. So there is no such $f$ and $\vec F$ is not conservative.

To find the work, first parameterize the path (call it $C$) by

$\vec r(t)=(1-t)(0,0,0)+t(-4,5,-5)=(-4t,5t,-5t)$

for $0\le t\le1$. Then

$\vec r'(t)=(-4,5,-5)$

and the work is given by the line integral,

$W=\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\int_0^1(e^{-4t}+5t,e^{-4t}+5t,-(e^{-4t}+5t)5t)\cdot(-4,5,-5)\,\mathrm dt$

$W=\displaystyle\int_0^1(125t^2+5t+(25t+1)e^{-4t})\,\mathrm dt=\boxed{\frac{2207}{48}-\frac{129}{16e^4}}$