Let C be the boundary of the region in the first quadrant bounded by the x-axis, a quarter-circle with radius 3, and the y-axis, oriented counterclockwise starting from the origin. Label the edges of the boundary as $C_1$,$C_2$,$C_3$ starting from the bottom edge going counterclockwise. Give each edge a constant speed parametrization with domain $0\leq t\leq 1$ :

edge $C_1$
$x_1(t)=$
$y_1(t)=$

edge $C_2$
$x_2(t)=$
$y_2(t)=$

edge $C_3$
$x_3(t)=$
$y_3(t)=$

$\displaystyle\oint\limits_C y^2x\,dx+x^2y\,dy$ $\displaystyle{}=\int\limits_{C_1} y^2x\,dx+x^2y\,dy$ $\displaystyle {}+\int\limits_{C_2} y^2x\,dx+x^2y\,dy$ $\displaystyle{}+\int\limits_{C_3} y^2x\,dx+x^2y\,dy$
$\hspace{10pt} =$ + +
Applying Green's theorem,

$\displaystyle\oint\limits_C y^2x\,dx+x^2y\,dy =$

$\displaystyle\int$

$\displaystyle\int$

$dx\,dy$

=

$\displaystyle\int$

$dy=$

The vector field $F=y^2x\,\hat{\imath}+x^2y\,\hat{\jmath}$ is: conservative not conservative