Explaining result: comparing line integral and two-form integral

Let $S$ be the hemisphere $x^2 + y^2 + z^2 = a^2$ , $z \geq 0$ , oriented with unit normal pointing upward. Let $C$ be the boundary curve, $x^2 + y^2 = a^2$ , $z= 0$ , oriented counterclockwise. Calculate

(a) $\int_{S}{dx \wedge dy + 2zdz \wedge dx}$
(b) $\int_{C}{xdy + z^{2}dx}$

Compare your answers and explain.

Solution:

Using spherical coordinates, I got $\pi a^2$ for both parts. The only part I'm not sure about is the explanation. Is this an application of Green's Theorem?