Chapter 16: Integrals and Vector Fields - Section 16.4 - Green's Theorem in the Plane - Exercises 16.4 - Page 979: 22

$-2$

The tangential form for Green Theorem -- Counterclockwise Circulation $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy $ Now, $\oint_C F \cdot n ds= \iint_{R} \dfrac{\partial (2x)}{\partial x}-\dfrac{\partial 3y)}{\partial y} dx dy $ or, $= \int_{0}^1 \int_0^{1-x} 2-3x dx dy$ or, $= -\int_{0}^{\pi} \int_0^{\sin x} dy dx$ or, $= -\int_{0}^{\pi} \sin x dx$ or, $[\cos x]_0^{\pi}=-2$