Question

Use Green's theorem to evaluate the integral:
y^(2)dx+xy dy
where C is the boundary of the region lying between the graphs of y=0,
y=sqrt(x), and x=9

Answer 1

Using the definition, we have
P = y^2
Q = xy

and the integral becomes

∫[0,9]∫[0,√x] (∂Q/∂x - ∂P/∂y) dy dx
= ∫[0,9]∫[0,√x] (y - 2y) dy dx
= ∫[0,9]∫[0,√x] -y dy dx
= ∫[0,9] -x/2 dx
= -x^2/4 [0,9]
= -81/4