A Calculus Analogy: Integrals as Multiplication

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.

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/a-calculus-analogy-integrals-as-multiplication/

Awesome! As always :wink:

@ Yifeng: Thanks! :slight_smile:

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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.

[…] A Calculus Analogy: Integrals as Multiplication | BetterExplainedbetterexplained.com […]

i am an engineer. always struggled with the ‘intuitive’ understanding. THANKS A TON !

@bivi: You’re welcome, glad it was helpful for you!

[…] Today’s installment: Integration is just multiplication when one of the operands is changing. […]

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WOW! One of the best yet. Hope things are well. Dan F. says ‘hi’. He is back and now a Dr.


@Mr. Rose: Thanks! Tell Dan hi also, I should be coming by later this summer :). Looking forward to catching up.


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.

@Parand: I know what you mean – we only learn much later that all the formulas are connected (vs. memorizing 1/2 gt^2 as the time taken to fall).

With the right analogies, I think the principles of calculus can be introduced much earlier.

This is one of the best explanations for intuitive understanding of calculus that I have ever seen - and I am a Maths Major!

@NK: Thank you! I really want to make these topics clear, I’m glad it came through :slight_smile:

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.

So now I’m waiting for your piece on derivatives. :slight_smile:

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.

So now I’m waiting for your piece on derivatives. :slight_smile:

@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 :).