Use Green's Theorem to calculate the work done by the force $\overrightarrow { \mathbf { F } } \text { on a particle that is }$ moving counterclockwise around the closed path C.
$\begin{array} { l }
\overrightarrow { \mathbf { F } } ( x , y ) = 5 x y \hat { \mathbf { i } } + ( x + y ) \hat { \mathbf { j } } \
C : x ^ { 2 } + y ^ { 2 } = 16
\end{array}$
A) $17 \pi$
B) 0
C) $16 \pi$
D) $3 \pi$
E) $15 \pi$

Question

Use Green's Theorem to calculate the work done by the force $\overrightarrow { \mathbf { F } } \text { on a particle that is }$ moving counterclockwise around the closed path C. 
$\begin{array} { l } 
\overrightarrow { \mathbf { F } } ( x , y ) = 5 x y \hat { \mathbf { i } } + ( x + y ) \hat { \mathbf { j } } \
C : x ^ { 2 } + y ^ { 2 } = 16
\end{array}$
A) $17 \pi$
B) 0
C) $16 \pi$
D) $3 \pi$
E) $15 \pi$

Quizplus · Accepted Answer

Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. For a vector field $\overrightarrow{\mathbf{F}} = M\hat{\mathbf{i}} + N\hat{\mathbf{j}}$, the work done around C is given by the line integral $\oint_C \overrightarrow{\mathbf{F}} \cdot d\overrightarrow{\mathbf{r}}$, which, according to Green's Theorem, equals the double integral over D of $(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y})dA$.Given $\overrightarrow{\mathbf{F}} = 5xy\hat{\mathbf{i}} + (x + y)\hat{\mathbf{j}}$, we have $M = 5xy$ and $N = x + y$. Thus, $\frac{\partial N}{\partial x} = 1$ and $\frac{\partial M}{\partial y} = 5x$. The integrand becomes $1 - 5x$.The region D is a circle with radius 4 (since $x^2 + y^2 = 16$), so we convert to polar coordinates where $x = r\cos(\theta)$ and $y = r\sin(\theta)$, and $dA = r dr d\theta$. The integral simplifies to integrating over the circle, which means $r$ goes from 0 to 4 and $\theta$ goes from 0 to $2\pi$.However, in this specific case, the correct calculation involves recognizing that the integrand simplifies correctly to an expression that, when integrated over the circle, yields $16\pi$, indicating a calculation error in the explanation process. The correct approach involves integrating the difference $(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y})$ over the area of the circle, taking into account the correct limits and the nature of the vector field.

Use Green's Theorem to Calculate the Work Done by the Force