Evaluate $\iiint xyz$ 
Evaluate
  $$\iiint_E xyz\, dV$$
  where $E$ is the solid: $0\leq z\leq 9,\,0\leq y\leq z,\, 0\leq x \leq y.$

I am having a hard time drawing a picture of this solid $E$ to find out what the limits of integration can be. I see when $x=0$ the solid in the $yz$-plane is a triangle and then as we move down the positive $x$-axis for $0\leq x \leq 9$, the triangle gets smaller and converges to a point when $x=9$. That is about all I have. I don't know how to visualize this perfectly enough in my mind and hold it steady to see which direction I should integrate in first and then find out what are the equations of the planes that I need to set up the integral. Any help is greatly appreciated.
 A: You can find the limits of integration by solving the inequalities. But in this case, the inequalities are already in solved form, if you integrate in the order $x,y,z$ (listed from innermost to outermost), so there's nothing to do.

If you wanted to integrate in the order $z,y,x$, we could solve in this other direction. In this case, it's pretty easy: we can combine all of the inequalities into one
$$ 0 \leq x \leq y \leq z \leq 9 $$
and then just break it apart in the other direction:
$$ 0 \leq x \leq 9 \qquad x \leq y \leq 9 \qquad y \leq z \leq 9 $$
Or if we wanted $x,z,y$, we could use
$$ 0 \leq y \leq 9 \qquad 0 \leq x \leq y \qquad y \leq z \leq 9 $$
This last version is neat, because the two inner integrals over $x$ and $z$ are actually independent of each other, and you can save yourself a bit of trouble by factoring the double integral into a product of two single integrals.
It's actually okay to get things too wide as you're working, as long as you don't throw away information in your inequalities: you'll correct yourself later on.
If, in the last example, I determined that $y$ was restricted to $0 \leq y \leq 10$, then later on when I moved to solve for $z$, I realize I need to split $y \leq z \leq 9$ into the $y \leq 9$ and $y \geq 9$ cases in order to properly write down bounds on my integrals and will quickly realize that there is nothing to integrate in the $9 \leq y \leq 10$ region. (this was a little obvious in this case, but the actual extents of your variables more complicated system of inequalities can take more effort to pin down exactly)
A: 
You might think about a diagram like this:
region between the white planes for x,
region between the pink planes for y,
region between the cyan planes for z.
The solid is a bit easier to see now, demarcated with blue points
