Use Green's Theorem to find the work done by the force $$\mathbf { F } ( x , y ) = \left( 6 x - 7 y ^ { 2 } \right) \mathbf { i } + 3 y \mathbf { j }$$
in moving a particle in the positive direction once around the triangle with vertices $( 0,0 ) , ( 1,0 )$, and $( 0,1 )$.
A) 7
B) $\frac { 7 } { 6 }$
C) $\frac { 7 } { 3 }$
D) 42

Question

Quizplus · Accepted Answer

Green's Theorem relates a line integral around a simple closed curve to a double integral over the region it encloses. The work done is the line integral of $\mathbf{F}$ around the triangle, which equals the double integral of $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}$ over the triangular region. Here, $\frac{\partial N}{\partial x} = 0$ and $\frac{\partial M}{\partial y} = -14y$. The integral over the triangular region gives $\frac{7}{3}$.

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