Parameterize the surface of interest [tex]S[/tex] by [tex]\mathbf x(u,v)=\langle9u\cos v,10u\sin v,10-90u\cos v-30u\sin v\rangle[/tex] with [tex]u\in[0,1][/tex] and [tex]v\in[0,2\pi][/tex].
Then the area is given by the surface integral
[tex]\displaystyle\iint_S\mathrm dA=\int_0^1\int_0^{2\pi}\left\|\frac{\partial\mathbf x}{\partial u}\times\frac{\partial\mathbf x}{\partial v}\right\|\,\mathrm dv\,\mathrm du=90\sqrt{110}\pi[/tex]
