Evaluate the triple integral, tetrahedron $$\iiint_E x^2dV, \text{where E is the solid tetrahedron with vertices }(0,0,0), (1,0,0), (0,1,0), \text{and (0,0,1)}$$
I need some assistance on setting up the limits. If someone could help me learn how to set up my limits of integration, that would be great. I should be able to integrate it just fine. 
 A: Your tetrahedron is bordered by the coordinate planes, and the plane $x+y+z=1$. Sketch a picture! So $x\ge0,$ $y\ge0$, $z\ge0$ and $x+y+z\le1$. 
[Edit: If you have trouble with this step, then go and review how you determine
the equation of a plane given that you know three points on it. Here the coordinate planes $x=0$, $y=0$ and $z=0$ stand out. The fourth plane passes via the points $(1,0,0), (0,1,0)$ and $(0,0,1)$. If everything else fails, you can go through that process. Here with a bit of experience you should notice that the coordinates of these three points sum up to $1$. That gives you the equation of the last plane./Edit]
As a next step you should ask yourself the questions:


*

*If I know the values of $x$ and $y$, what is the allowable range for $z$?

*Ignoring $z$, if I know the value of $x$, what is the allowable range for $y$?

*Ignoring $z$ and $y$, what is the allowable range for $x$?


This will give you the limits, if you first integrate w.r.t. $z$, then $y$, last $x$.
If you prefer (sometimes it will be to your advantage), you can process the variables in a different order.
A: Your volume is bounded by the plane $x+y+z=1$ and the coordinate axes.  Therefore, one way to express this integral is as follows:
$$\int_0^1 dx \, x^2 \, \int_0^{1-x} dy \, \int_0^{1-x-y} dz $$ 
The way to interpret this integral is to imagine a single point in the $xy$ plane, and a line through that point that varies from $z=0$ to where it hits the plane $x+y+z=1$, at $z=1-x-y$.  The inner integral represents integrating the function over that line in $z$.  You then integrate over all points $(x,y)$ within the tetrahedron, and we choose all $y$ between $y=0$ and $y=1-x$, and $x$ between $x=0$ and $x=1$.
You could easily evaluate the integral in a different order, viz.
$$\int_0^1 dy  \, \int_0^{1-y} dz \, \int_0^{1-z-y} dx \, x^2 $$ 
although, given the function you want to integrate over this tetrahedron, it may be easier to integrate in one arrangement than in another.
