Calculate the surface integral \iint_M (\nabla \times \mathbf{F}) \cdot d\mathbf{S} where $M$ is the hemisphere $x^2 + y^2 + z^2 = 4$, $x \ge 0$, with the normal in the direction of the positive x direction, and \mathbf{F} = (x^4, 0, y^2).
Begin by writing down the \"standard\" parametrization of $\partial M$ as a function of the angle $\theta$ (denoted by \"t\" in your answer)
x =
y =
z =
$\int_{\partial M} \mathbf{F} \cdot d\mathbf{s} = \int_0^{2\pi} f(\theta) \, d\theta$, where
f(\theta) =
(use \"t\" for theta).
The value of the integral is
