Assuming[{l>0,lb>0,alpha>0,Y>0,R>0},Integrate[Pi*r*Y/l*(Sqrt[lb^2+r^2*alpha^2]-l)^2,{r,0,R}]]

%/.lb->l-1/4*R^2*alpha^2/l

Assuming[l>0,FullSimplify[%]]

%/.{R->l*eps}

Q=Assuming[l>0,FullSimplify[%]]

FullSimplify[Q]

Series[Q,{eps,0,500}]

Integrate[(Sqrt[x]-l)^2,x]

Assuming[{lb>0,R>0,alpha>0,dzeta>0},Integrate[Sqrt[1+(lb/(R*alpha)-dzeta/alpha*Cos[a])^2],{a,0,alpha}]]

Series[Sqrt[1+x],{x,0,4}]

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