I'm always wondering the answer to this question: Why does multiplication coupled with some sort of integral calculus work the way it does? We multiply to get

I'm always wondering the answer to this question: Why does multiplication coupled with some sort of integral calculus work the way it does? We multiply to get moments of a distribution, we multiply to convolve, we multiply to get the work done on an object. I suppose the answer is multiplication allows us to scale some function f(x) with some function g(x). But I guess I want something deeper and I feel like I'm missing it.