Signed areas

The lens-shaped area should remain the same if it is translated along the y-direction. For this to happen, we need the concept of negative areas.

f1[a_,x_]:=-x^2+1+a; (* the "top" function *)
f2[a_,x_]:=x^2-1+a; (* the "bottom" function*)

f3[a_,x_]:=f1[a,x]-f2[a,x];
integ[a_,x_]:=Integrate[f3[a,t],{t,-1,x}];

Manipulate[
Plot[{f2[a,x],f1[a,x],f3[a,x](*,integ[a,x]*)},{x,-1,1},Filling->Axis,PlotRange->4],{a,3,-3,-0.5}]