Repeated addition is just one multiplication algorithm. I think a much better way to think of multiplication is as an operation on numbers that is related to a

Repeated addition is just one multiplication algorithm. I think a much better way to think of multiplication is as an operation on numbers that is related to addition by the distributive law. That law gives the "multiplication is repeated addition" algorithm, as well as the "shift and add" algorithm that most people learn in grade school. Of course, there are more efficient multiplication algorithms than either of those, which is why it is better to think of multiplication independently from addition. I would also point out that the repeated addition algorithm makes a lot of sense in base 1, but that shift and add makes far more sense in base 2 or higher. As for integration, I would say it is better to think of it as a limit involving both addition and multiplication.