Surface and Volume Integrals - Limits of Integration

[Bruce Dawk]

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 can't 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 let's just treat this as a single integral and integrate over the hard limits of y.

 

[STEMucator]

x and y determine the shape of the surface, not the curve. You must determine the limits for x and y respectively depending on the region which Is given.

So each case will be different depending on the surfaces you are given to integrate over.

 

[marxLynx]

The only connection the region of integration has to the surface is that the region lies in the domain of the surface. The reason one would express a limit of x as a function of y is when one integrates a region that has a boundary that isn't constant at all its points (with respect to x). An example would be if the region is a right triangle or something. The hypotenuse of the triangle has some equation that describes it as a line with a restricted domain. When you integrate first in the x-direction, the equation tells the integral when to stop collecting x-elements ( or where to start). The equation of the line contains x's and y's only because it is within the same coordinate system. I'm not sure what you mean by "hard limits of x."