I thought about posting this in the math forums, but I'm curious about what integrals actually represent as applied to physics so I'm posting it here. I feel that others will conceptually benefit from this conversation as well. Differentiation has a clear and easily understandable meaning and application since there are rates everywhere in the physical world. We also all know that integration is simply the "opposite" of differentiation and it can be thought of as the area under a curve by summing the infinitely many rectangles of ever decreasing width. ... So that all makes sense, but can someone talk about how integration fits into the physical world? Derivatives are to rates as integrals are to... what?