A particle starts at the point $( - 3,0 )$, moves along the $x$-axis to $( 3,0 )$ and then along the semicircle $y = \sqrt { 9 - x ^ { 2 } }$ to the starting point. Use Green's Theorem to find the work done on this particle by the force field $\mathbf { F } ( x , y ) = \left\{ 24 x , 8 x ^ { 3 } + 24 x y ^ { 2 } \right\}$
Select the correct answer.
A) 486
B) $48 \pi$
C) 0
D) $486 \pi$
E) $972 \pi$

Question

Quizplus · Accepted Answer

We apply Green's Theorem,
\begin{align*}
\oint _ { C } \mathbf { F } \cdot d \mathbf { r } & = \iint _ { R } \left( \frac { \partial Q } { \partial x } - \frac { \partial P } { \partial y } ight) d x d y \
& = \iint _ { R } ( 24 y ^ { 2 } - 24 y ^ { 2 } ) d x d y \
& = \iint_R 0\,dxdy \
& = 0
\end{align*}
because the integrand is an odd function and the region is symmetric with respect to the $x$-axis. Therefore, the work done by the force field is zero. So, the correct choice is $\boxed{	extbf{(C)}}$.

A Particle Starts at the Point (−3,0)( - 3,0 )(−3,0) , Moves Along The

A Particle Starts at the Point $( - 3,0 )$ , Moves Along The