# Help specifying limits of integration on double integral

I am learning about the limits of integration for double integrals. One of the problems was, "Find the volume of the wedgelike solid that lies beneath the surface $$z = 16 - x - y$$ and above the region R bounded by the curve $$y = 2\sqrt{x}$$, the line $$y = 4x - 2$$, and the x-axis." Using vertical cross sections, (dy dx) I had my $$y$$ limits as: $$(4x-2,2\sqrt{x})$$ and my x limits as $$(0,1)$$; however, the book said this was not possible and that you would need 2 integrals instead because, "y varies from $$y = 0$$ to $$y = 2\sqrt{x}$$ for $$0<=x<=0.5$$, and then varies from $$y = 4x - 2$$ to $$y = 2\sqrt{x}$$ for $$0.5 <= x <= 1$$." They then went in order of dxdy instead. What does it mean for y to "vary" on those intervals?

• To @user604720 : Check my answer, I believe it is easy now to understand the meaning for $y$ to "vary"... – Anton Vrdoljak Apr 13 at 22:15

I would draw region $$R$$, here is image done by GeoGebra: 