Parameterize [tex]\mathcal S[/tex] by
[tex]\mathbf s(u,v)=\langle4\cos u\sin v,4\sin u\sin v,4\cos v\rangle[/tex]
with [tex]0\le u\le2\pi[/tex] and [tex]0\le v\le\dfrac\pi6[/tex]. Then
[tex]\displaystyle\iint_{\mathcal S}y^2\,\mathrm dS=\int_{u=0}^{u=2\pi}\int_{v=0}^{v=\pi/6}(4\sin u\sin v)^2\|\mathbf s_u\times\mathbf s_v\|\,\mathrm dv\,\mathrm du[/tex]
[tex]=\displaystyle\int_{u=0}^{u=2\pi}\int_{v=0}^{v=\pi/6}16\sin^2u\sin^2v\times16\sin v\,\mathrm dv\,\mathrm du[/tex]
[tex]=\displaystyle256\left(\int_{u=0}^{u=2\pi}\sin^2u\,\mathrm du\right)\left(\int_{v=0}^{v=\pi/6}\sin^3v\,\mathrm dv\right)=\frac{32(16-9\sqrt3)\pi}3[/tex]
