$\displaystyle F.dr=\ \left(3x^2-8y^2\right)dx+\left(4y-6xy\right)dy \\[2ex] \displaystyle let\ \ P=3x^2-8y^2\ ,\ \ Q=\left(4y-6xy\right)\ \\[2ex] \displaystyle \frac{\partial{}P}{\partial{}y}=\ -16y,\ \ \frac{\partial{}Q}{\partial{}x}=-6y\ \ \\[4ex] \displaystyle \ Green’s\ Theorem,\\[4ex] \displaystyle \int_cPdx+Qdy=\iint_R\left(\frac{\partial{}Q}{\partial{}x}-\frac{\partial{}P}{\partial{}y}\right)dx\ dy \\[4ex] \displaystyle \iint_R-6y-\left(-16y\right)dx\ dy=10\iint_Ry\ dx\ dy \\[2ex] $

$\displaystyle x=0\ to\ \ x=1-y\ \\[2ex] \displaystyle y=0\ to\ 1\ \ \\[2ex] \displaystyle 10\int_0^1\begin{array}{l}\int_0^{1-y}ydxdy \\ \ \end{array}=\ 10\int_0^1y{\left[x\right]}_0^{1-y}dy\\[3ex] \displaystyle =10\int_0^1y\left(1-y\right)dy\ \\[2ex] $

$\displaystyle 10\ {\left[\frac{y^2}{2}-\frac{y^3}{3}\right]}_0^1=10\ \left[\frac{1}{2}-\frac{1}{3}\right]=\frac{10}{6}\ \\[3ex] \displaystyle \iint_R\left(\frac{\partial{}Q}{\partial{}x}-\frac{\partial{}P}{\partial{}y}\right)dx\ dy=\frac{10}{6}=\frac{5}{3}\ …\left(A\right) \\[2ex] $

$\displaystyle part\ 1\ :Along\ \ x+y=1 \\[2ex] \displaystyle dx+dy=0\ ,\ dy\ =\ -dx \\[2ex] \displaystyle y=1-x \\[2ex] \displaystyle \int_1^0\left(3x^2-8{\left(1-x\right)}^2\right)dx\ +\left(4\left(1-x\right)-6x\left(1-x\right)\right)\left(-dx\right) \\[2ex] \displaystyle -\int_0^1\left(3x^2-8-8x^2+16x+4x-4+6x-6x^2\right)dx\\[2ex] \displaystyle =\ -\int_0^1-11x^2+26x-12\ dx\ \\[2ex] \displaystyle =-{\left[-\frac{11x^3}{3}+\frac{26x^2}{2}-12x\right]}_0^1\\[2ex]=\ \dfrac{11}{3}-13+12\\[2ex]=\dfrac{8}{3}\ \\[2ex] $

$\displaystyle part\ 2\ :along\ x\ axis,\ y=0,\ dy=0,\ \\[2ex] \displaystyle \int_0^13x^2dx={\left[x^3\right]}_0^1=1\ \\[2ex] $

$\displaystyle part\ 3:along\ y\ axis,\ x=0,\ dx=0 \\[2ex] \displaystyle \int_1^04y\ dy=4\ {\left[\frac{y^2}{2}\right]}_1^0=2\left[-1\right]=\ -2 \\[2ex] $

$\displaystyle hence,\ \ \ \int_c\left(3x^2-8y^2\right)dx+\left(4y-6xy\right)dy=\ \frac{8}{3}+1-2=\frac{5}{3}…(B) \\[4ex] \displaystyle From\ \left(A\right)and\ \left(B\right),\ \\[4ex] Green\ Theorem\ is\ verified.\ \\[2ex] $