So I am trying to understand how and why the limits of surface and volume integrals come about. I think I came up with a easy to understand argument but not a mathematically sound one. Frankly its a little dodgy. Can anyone provide feedback on this argument or provide a better and possibly more mathematically sound explanation? In single integrals where y = f(x) we do ∫f(x)dx, this is because it is solely the x values that draw out the curve that we are interested in so the limits are based on the values of x that we are interested in. In double integrals for the most general case both x and y determine the shape of the curve, as a consequence we cant simply just integrate over x by placing the hard limits of x, rather we express the x limits as functions of y as we accept that they are related and once we work out the x integral we say well, we have taken into account the dependency of x that y has, so now lets just treat this as a single integral and integrate over the hard limits of y.