Integrals are often described as finding the "area under the curve". This description is misleading, like saying multiplication is for finding "the area of a rectangle". Finding area is a useful application, but not the purpose. Integrals help us combine numbers when multiplication can't.

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You might be interested to know that physicists (as opposed to mathematicians) often explain integration in more or less just this way. Physicists tend to be less rigorous than mathematicians in their use of math, but the goal is to understand things intuitively.

@Mitch: Thanks for the comment! Yes, I think in physics it starts to become clear that integration is closely linked to multiplication in some way.

Unfortunately, calculus is often taught in the absence of physics, so students often don’t see these analogues and it remains a strange operation. Physics definitely helped me understand calculus at a more intuitive level.

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I remember being absolutely incensed when I finally learned calculus in high school. All the formulae I’d had to memorize in physics and chemistry finally made sense. Why not teach calculus first and save me all that rote memorization?

I think calculus as a concept is simpler than people assume it to be. A general, perhaps not heavily mathematical introduction very early on would be tremendously valuable in framing and helping understand the sciences.

This was an incredible explanation. Integrals never made any intuitive sense to me before, and existed only as a series of mathematical steps divorced from any real world meaning.

This was an incredible explanation. Integrals never made any intuitive sense to me before, and existed only as a series of mathematical steps divorced from any real world meaning.

@Bryan: Awesome, glad you liked it! i know what you mean, it took a long time for me to start seeing integrals as something beyond a formal mathemetical step.

I’m looking forward to writing on derivatives too :).