Can double integrals be interpreted as net change?

[Emekadavid]

I understand that single integrals over a function can be interpreted as net change. Net change of the quantity between the bounds of the integration. But I am trying hard to understand if double integration can also be regarded as net change? That is, the net change in volume when the two input variables are changing at the same time, or the net change when on variable is held constant and the other variable to the function is changing? I am interpreting it this way because of dA in double integral acting over the function, f(x,y). Am I getting it wrong and can someone clarify me on the intuitive understanding? The textbook I am reading on double integrals does not state it as such but it uses limits of Riemann's sums to prove double integrals as an approximation of the volume so that is why I am trying to see if there is a connection?
thanks

 

[jvicens]

I can clarify the intuitive understanding of double integrals for you. Double integration can indeed be interpreted as net change, but in this case, it represents the net change in volume. This is because double integrals are used to find the volume under a surface in three-dimensional space.

To better understand this, let's first consider the single integral. When we integrate a function over a certain interval, we are essentially finding the net change in that function over that interval. For example, if we have a function that represents the velocity of an object, integrating it over a certain time interval would give us the net change in the position of the object over that time interval.

Similarly, in double integration, we are finding the net change in volume under a surface. This can be thought of as the net change in the quantity represented by the surface, such as the amount of water in a container or the amount of air in a room.

To answer your question about whether double integration can be regarded as net change when one variable is held constant and the other is changing, the answer is yes. This is because in this case, we are essentially looking at the net change in volume as one variable changes, while the other remains constant.

The use of dA in double integrals represents the infinitesimal area of the surface, which is then multiplied by the function f(x,y) to find the volume under that small area. By summing up all these infinitesimal volumes, we can approximate the total volume under the surface.

I hope this clarifies the intuitive understanding of double integrals for you. Remember, in the end, it all comes down to finding the net change in volume under a surface, whether both variables are changing or one is held constant.