I'm working through McDuff and Salamon's `Introduction to Symplectic Topology' (highly recommend). In example 3.1.2, we are given the example of a symplectic manifold: the 2-sphere with its standard area form. Spcifically, $$S^2 = \{(x_1,x_2,x_3)\in\mathbb{R}^{3}\:|\: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}$$ with induced area form $\omega_{x}(\xi,\eta) = \langle x,\xi\times\eta\rangle$ for $\xi,\eta\in T_{x}S^{2}$. The authors then comment that the total area is $4\pi$.
So my question is: what is the easiest way to see that the total area is $4\pi$? I would like to know how I can integrate this 2-form $$\int_{S^{2}}\omega$$ possibly using Stoke's theorem. I would also have a preference for coordinate free solutions!
Thanks in advance for your help.
If you read further into the book you'll learn many coordinate-free ways of seeing this, but they're not "directly" calculating the integral and depend on some extra structure that the sphere has. To do it directly, you do need coordinates.
The easiest coordinates in this case are cylindrical ones, you can check that writing the form in those terms gives $dz\wedge d\theta$, now the integral is easy to take.