# Plot a region between two curves that is below another in $R^3$

I am trying to obtain a 3D plot of $$z=x^2+2y^2$$ that is bound by the region in the $$z=0$$ plane governed by the equations $$y=x$$ and $$y=x^2$$. Is it possible to do this with RegionFunction? or RegionPlot3D?

Plot3D[