Why does it matter whether we use dx or dy (Disk and Washer)? In solving for volumes, why do we use dx in some cases and dy in other cases? If both quantities are approaching 0, then why does it matter which one we use? I understand we can tell which one to use by the axis that the cuts are being made, and that we can replace using the derivative, but why does this actually work? In other words, can someone plz provide some sort of intuition for what an integral involving two different variables means? 
Thanks,
Dude156
 A: A volume can be expressed as an area times a height. $dxdy$ is an area element, because it is an infinitesimally small area where $dx$ is the length and $dy$ is the width (or vice-versa). When you have a function $z = f(x,y)$ that can be interpreted as the "height" of the body at a given point $(x,y)$, by multiplying it by $dxdy$ you obtain a volume element. By summing (integrating) all the volume elements within a body, you obtain the total value of the volume.
This is analogous to the area under a curve in the case of the one-variable integral. The function $f(x)$ is the "height", and $dx$ is an element of length, which constitutes the "base" of the area element. When you sum all area elements $f(x)dx$, that is, when integrating the function over a given interval, you obtain the total area under the curve.
When you mutiply a density ($\frac{mass}{volume}$) by a volume, you obtain the total mass. So when you integrate a density function $\delta(x,y,z)$ over a given volume by means of a triple integral, you are simply multiplying each volume element (base * width * height or $dxdydz$) by the density at that point, and adding all the little mass elements that you obtain, which ends up giving the total mass of a body.
You must be careful to use $dx$, $dy$ and any other differential properly, because they indicate which variable is changing over a summation (a given integral) and which others remain constant. 
