Question

Verify Green’s theorem in the plane for $ (xy + Sy?)dx + 3xdy,

where c is the closed curve of the region bounded by y = x and y = x?.

Answer 1

Green’s theorem:

$\oint_CPdx+Qdy=\iint_D(Q_x-P_y)dA$

We have:

$P=xy+y,Q=3x$

Then:

$Q_x=3,P_y=x+1$

$\oint_C(xy+y)dx+3xdy=\int^1_0(2-x)dx\int^{1-x}_xdy=\int^1_0(2-x)(1-2x)dx=$

$=(2x-5x^2/2+2x^3/3)|^1_0=2-5/2+2/3=1/6$

$\oint_C(xy+y)dx+3xdy=\int_1^{0.5}[(x(1-x)+1-x)dx+3xd(1-x)]+$

$+\int_{0.5}^{0}[(x^2+x)dx+3xdx]=\intop^{0.5}_1(1-x^2-3x)dx+\int^0_{0.5}(x^2+4x)dx=$

$=(x-x^3/3-3x^2/2)|^{0.5}_1+(x^3/3+2x^2)|^0_{0.5}=$

$=0.25-0.125/3-0.75/2-1+1/3+1.5-0.125/3-0.5=1/4-1/12-3/8+1/3=1/6$

We have the same results, so Green’s theorem is verified.