Question

if the first derivative is area, and the second intergral is volume, what does the third integral give as an answer?

Answer 1

When setting up multiple integrals you know if you want the area you do "top curve minus bottom" or "right minus left" depending on with respect to what variable.
However, area is typically done using a double integral in which you don't have to solve the function explicitly in terms of one variable because you can let x and y vary as they see fit.
When you set up to find the volume using 2 integrals. You do "top surface minus bottom surface". However, again, volume is best formulated using a triple integral where you can let x,y, and z vary. Typically you can assume the surface is in the first octant (or region in the first quadrant) and then you get top curve-0 (also, top surface-0) so the z variance is disregarded.