I would like to know if there's a way to rearrange the order of integration without drawing a picture.
For example, suppose I have: $\int_{0}^{4} \int_{0}^{4-y} \int_{0}^{\sqrt{z}} dxdzdy$
If I want to rearrange this so that it is in the order $dzdydx$, how would I go about doing this algebraically?
You have that $$0\leq y\leq 4,\quad 0\leq z\leq 4-y,\quad 0\leq x\leq \sqrt z,$$
So you have that $y$ is independent, $z$ depending on $y$ only and $x$ depending on $y$ and $z$. You can also write this expression as :
$$0\leq x\leq \sqrt{z}\leq \sqrt{4-y}\leq 2 \quad \text{or}\quad 0\leq x^2\leq z\leq 4-y\leq 4.$$
What you want is $x$ independent, $y$ depending on $x$ only and $z$ depending on $x$ and $y$. So take $x\in [0,2]$ and $x\leq \sqrt{4-y}\leq 2$, i.e. $0\leq y\leq 4-x^2$, and finally $x^2\leq z\leq 4-y.$
To understand better, make a dray (even if in 3D it's more complicate)...