Here's an example from the Stewart text that I'm struggling with. [For those with the text this is on page 1074 from the 8th edition].
Suppose we are given $\int_0^1\int_0^{x^2}\int_0^y f(x,y,z) dzdydx$ and we want to rewrite this iterated integral in other ways.
First the three dimensional region (as given) is clearly
$$E = \{(x,y,z)| \quad 0\le x\le 1\quad 0\le y\le x^2\quad 0\le z\le y\}$$
Then the various different representations are
on the xy-plane $$D_1 = \{(x,y) | 0\le x\le 1; 0\le y\le x^2\}$$ $$D_1 = \{(x,y)| 0\le y\le 1;\sqrt{y}\le x\le 1\}$$ $$D_2 = \{(y,z)|0\le y\le1, 0\le z\le y\}$$ $$D_3 = \{(x,z)|0\le x\le 1, 0\le z \le x^2\}$$
I thought that the two representations for $D_1$ were pretty clear since if $z=0$ then it is still the case that $0\le x\le 1$ and $0\le y\le x^2$ and since the xy plane is precisely when $z=0$ then $D_1$ follows quite naturally. I don't get the same easy feeling with $D_2$ and $D_3$.
My thought was that because I want to project $E$ onto the yz for $D_2$ then I ought to set $x=0$; then $0\le y\le x^2$ becomes $y=0$. That doesn't seem very helpful.... I'd appreciate help on this because I think this is the least intuitive concept I've seen in all of calculus. Thank you!