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I suppose that this differs at how mathematically mature the student is. For undergrads and high school students it is, as you say, probably a good idea to give a lot of motivation and applied examples for the theory being taught. I suppose this article is more geared towards teaching university students, and then it can well be that examples and applications become more abstract. Of course, introductory topics such as Linear algebra and Calculus would lend themselves very well for a wide range of application examples. On the other hand, in more advanced and abstract university courses the 'realistic problems' should already be reasonably clear for the student when starting the course.

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I'm making an assumption here wrt OP, but I think that a lot of people are taught that an integral is a set of fancy tricks that you use to simplify down equations until you get ... an answer.

In my experience, students are expected to memorize all sorts of equations without ever really being expected to understand why you're calculating the integral in the first place (even at decent universities). Obviously the risk with that is that you won't recognize any problem can be solved with integrals, no matter how concrete. Once I left the fancy tricks behind I felt like a fog lifted and it was actually really fun to see how integrals fit into both applied and abstract examples.

Most classes actually teaches why integrals are useful, the problem isn't the material but that student's quickly forget that and then only remembers that they did a lot of memorizations of formulas and solving algebra. Then once you have worked a lot with integrals you've built the intuition to actually understand what it does and then it seems like a simple explanation taught you everything, although most likely you already heard that simple explanation a long time ago and forgot about it since you didn't understand it at the time.

Fair point. Though I do think the focus on integration shortcuts was a huge distraction to the main content, and arguably irrelevant to the actual concepts.

My class consisted of hour after hour of solving integrals of different types using all the different shortcuts, and the exams were largely the same. I feel like that time and effort could have been much better spent.

In this case you have to bring them back to reality once in a while (like: "This is what we are actually doing here"). Ofc there are gifted math geniuses which have no problem with everything being abstract, but for most people connecting what they learn to the world they live in is a prerequisite for maintaining at least a minimum level of interest for the topic. At school students are used to learning useless stuff without questioning it. They won't tell you if what you teach them is useless stuff they don't understand.

Going back to that simple explanation once people are a bit more experienced can make things click for a lot of people. Remaining people what they are dealing with and why can also help.

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