In single integration, we essentially added all of the y-values for every x-value in a set domain. Back then, the definition of an integral on [a,b] was the limit of a Riemann sum:
In double integration, we do something where we split a region up into many squares, and we sum up the volumes of the prisms made from the squares. Now, we add up all of the z-values for every square created by the point (x,y).
To actually calculate a double integral, the area of the partitions approaches zero.
Before we start, we have to define the two types of regions.
This is "normal", or at least more familiar. This is when the y's are bound by two functions f(x) and g(x) and the x's are bound between constants.
This is inverted. This means the x's are bound by two functions f(y) and g(y), and the y's are bound between constants.