Yet, the idea of infinitesimals is very popular in applications. For example, in undergrad physics courses, when deriving formulas (essentially, doing math!) they keep talking about a "small area" or a "small volume" (where a given function is assumed to stay constant).

That's the beauty of mathematics - two people are able to approach a question from totally different perspectives, and produce the same result.

When you're actually computing things, such as numerically approximating an integral, you do so by breaking it into small pieces and adding them together. Infinitesimals are the limit of that process as the smallness goes to zero. I think the key problem here is that students use the symbolic rules but never actually do numerical computations of this kind.

I agree, I was a math major and remember how high school calculus using infinitesimals ended up confusing my friends in college.