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?

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

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

enter image description here

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