Parameterize [tex]C[/tex] by
[tex]\vec r(t)=\langle x(t),y(t)\rangle=\langle4\cos t,4\sin t\rangle[/tex]
with [tex]0\le t\le2\pi[/tex]. Then
[tex]\displaystyle\int_Cy^2\,\mathrm dx+x\,\mathrm dy=\int_0^{2\pi}\langle y(t)^2,x(t)\rangle\cdot\left\langle\frac{\mathrm dx(t)}{\mathrm dt},\frac{\mathrm dy(t)}{\mathrm dt}\right\rangle\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^{2\pi}\langle16\sin^2t,4\cos t\rangle\cdot\langle-4\sin t,4\cos t\rangle\,\mathrm dt[/tex]
[tex]=\displaystyle16\int_0^{2\pi}(\cos^2t-4\sin^3t)\,\mathrm dt=\boxed{16\pi}[/tex]
