Question

Use Green’s theorem to evaluate ∫ (x^5+2y)dx + (4x-6y^3)dy where C is the circle x^2 + y ^2 = 4 .

Answer 1

ANSWER¨ 8π

EXPLANATION.Let $D=\left\{ \left( x,y \right) :\quad { x }^{ 2 }+{ y }^{ 2 }\le 4 \right\}$ . The area of the D is 4π=$\iint_{D} dxdy$ By the Green's theorem $\oint { \left( { x }^{ 5 }+2y \right) dx+\left( 4x-6{ y }^{ 3 } \right) dy\quad } =$ $\iint_{D} \left [ \frac{\partial (4x-6y^{3})}{\partial x } -\frac{\partial(x^{5}+2y) }{\partial y}\right ]dxdy$ =$\iint_{D} \left [ 4-2\right ]dxdy$ =8π.