Hi, thanks for the comment. My main point is that going beyond the "area" understanding leads to a better understanding of calculus:
* Finding area does not readily imply an inverse operation, but multiplication does ("division" is differentiation, and it is the inverse). Most students would not posit the existence of an inverse of "finding the area", but multiplication is easy to reason about.
* Multiplication scales to N variables (multiple integration), while the mental model of "area" breaks down after the 3rd dimension.
* Multiplication matches more closely to what is meant in a lot of cases. When you integrate speed and time, you are trying to multiply them. You probably don't mean to imply that you're going to plot them on a graph and take the area under the curve as distance. That may be the mechanics of it, but finding area is not the purpose of the operation.
So, area isn't wrong by any means, it's just a limiting viewpoint in my mind, and there are better analogies out there. Thanks for the comment!
* Finding area does not readily imply an inverse operation, but multiplication does ("division" is differentiation, and it is the inverse). Most students would not posit the existence of an inverse of "finding the area", but multiplication is easy to reason about.
* Multiplication scales to N variables (multiple integration), while the mental model of "area" breaks down after the 3rd dimension.
* Multiplication matches more closely to what is meant in a lot of cases. When you integrate speed and time, you are trying to multiply them. You probably don't mean to imply that you're going to plot them on a graph and take the area under the curve as distance. That may be the mechanics of it, but finding area is not the purpose of the operation.
So, area isn't wrong by any means, it's just a limiting viewpoint in my mind, and there are better analogies out there. Thanks for the comment!