Stokes' theorem says the integral of the curl of [tex]\vec F[/tex] over a surface [tex]S[/tex] with boundary [tex]C[/tex] is equal to the integral of [tex]\vec F[/tex] along the boundary. In other words, the flux of the curl of the vector field is equal to the circulation of the field, such that
[tex]\displaystyle\iint_S\nabla\times\vec F\cdot\mathrm d\vec S=\int_C\vec F\cdot\mathrm d\vec r[/tex]
We have
[tex]\vec F(x,y,z)=x^2\,\vec\imath+4x\,\vec\jmath+z^2\,\vec k[/tex]
[tex]\implies\nabla\times\vec F(x,y,z)=4\,\vec k[/tex]
Parameterize the ellipse [tex]S[/tex] by
[tex]\vec s(u,v)=\dfrac{u\cos v}{\sqrt5}\,\vec\imath+\dfrac{u\sqrt5\sin v}4\,\vec\jmath[/tex]
with [tex]0\le u\le1[/tex] and [tex]0\le v\le2\pi[/tex].
Take the normal vector to [tex]S[/tex] to be
[tex]\dfrac{\partial\vec s}{\partial\vec u}\times\dfrac{\partial\vec s}{\partial\vec v}=\dfrac u4\,\vec k[/tex]
Then the flux of the curl is
[tex]\displaystyle\iint_S4\,\vec k\cdot\dfrac u4\,\vec k\,\mathrm dA=\int_0^{2\pi}\int_0^1u\,\mathrm du\,\mathrm dv=\boxed{\pi}[/tex]
