p[u_,v_]={(a+b*Cos[u])Cos[v],(a+b*Cos[u])Sin[v], b*Sin[u]}

ParametricPlot3D[p[u,v]/.{a->4,b->1},{u,0,2Pi},{v,0,2Pi}]

dp[u_,v_]=d[p[u,v]]/.{d[a]->0,d[b]->0}

Simplify[Dot[dp[u,v],dp[u,v]]]

EE[u_,v_]=b^2

FF[u_,v_]=0

GG[u_,v_]=(((a+b Cos[u]))^2)

f[u_,v_]=Sqrt[EE[u,v]*GG[u,v]-(FF[u,v])^2]

Integrate[b*(a+b Cos[u]),{u,0,2Pi},{v,0,2Pi}]