Summer of Math Exposition

Nice submission! A few points of note:

Think about your target audience, Are they likely to be incredibly fluent with concepts like sigma notation, arrows as limits, R representing the real numbers, arrows as implications (especially if they’re struggling to understand integration)? It may be more clear to use words rather than inline expressions like these for an a-level student.

I like that you connect this back to the real world with the the water and distance analogies! It may help to state that the x axis would measure time to help a student visually connect back to the graph in the water analogy.

I like your diagrams to visualise the areas we care about for a definite integral, I do think the repeating vertical lines maybe are not a particularly clear way of presenting this though. A natural student question that goes unanswered in this pdf is “what would the area up to -10 look like” It’s not obvious how you diagram might handle a case where one limit is negative and one is positive. They might also ask “how do we think about area when a curve goes both above and below the x axis”?

More minor nitpicks: Be careful about your implications! In the definite integral section, you write $f(x)=x^2\iff F(x)=x^3/3$, but this is not true! this is a one way implication unless you add your constant of integration. In fact, it may make more sense to talk about indefinite integrals before definite integrals. I think it makes F(x) and the +c cancellation make more sense. This may just be personal preference though.

You write $dx\rightarrow\infty$ instead of 0 in the first page.

There is repetition across the sentence describing delta x and the limit notation for the riemann sum. They are not written clearly to me.

It might be helpful to connect integration back to the riemann sum by doing an example integrating a simple function like f(x)=x^2 and showing that it agrees with the power rule?

NIT: think dx -> \infty should be dx -> 0 The visuals were well done and helpful. I think the writing should’ve tried to do some of the motivation before introducing definitions rather than introducing definitions and trying to motivate it after the fact

The article was very well written and gave a clear and well visualized introduction to integration. I liked the tank filing analogy. However, it lacked a hook or interesting tidbit to differentiate it from the standard treatment that can be found in textbooks.

Well written but otherwise very standard explanation of Riemann Integrals.