[tex]\displaystyle\iint_Sf(x,y,z)\,\mathrm dS[/tex]
Parameterize the part of the plane, [tex]S[/tex], we're interested by
[tex]\mathbf s(u,v)=\left(u\cos v,u\sin v,-\dfrac v2(\cos v+2\sin v)\right)[/tex]
with [tex]0\le u\le1[/tex] and [tex]0\le v\le2\pi[/tex]. Then we have surface element
[tex]\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|\,\mathrm du\,\mathrm dv[/tex]
[tex]\mathrm dS=\dfrac{3u}2\,\mathrm du\,\mathrm dv[/tex]
and so the surface integral becomes
[tex]\displaystyle\iint_Sf(x,y,z)\,\mathrm dS=\iint_Sf(\mathbf s(u,v))\|\mathbf s_u\times\mathbf s_v\|\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\frac32\int_{v=0}^{v=2\pi}\int_{u=0}^{u=1}\left(3u^2\cos v\sin v+\dfrac{u^2}4(\cos v+2\sin v)^2\right)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\frac3{16}\int_{v=0}^{v=2\pi}\int_{u=0}^{u=1}u^2(5-3\cos2v+16\sin2v)\,\mathrm du\,\mathrm dv=\frac{5\pi}8[/tex]
