Manipulate[
 Row[{Plot3D[{4 - c x^2 - c y^2, 0}, {x, -4, 4}, {y, -4, 4}, Axes -> False, Boxed -> False, Mesh -> False, 
   PlotStyle -> Directive[Opacity[0.5], Red], ImageSize -> 500, PlotRange -> {0, 4}] ,
 NIntegrate[4 r - c r^3, {r, 0, Sqrt[4/c]}, {t, 0, 2 Pi}]}], {c, 0.25, 2}]