Integrals are often described as finding the area under a curve. This description is too narrow: it's like saying multiplication exists to find the area of rectangles. Finding area is a useful *application*, but not the purpose of multiplication.

**Key insight: Integrals help us combine numbers when multiplication can't.**

I wish I had a minute with myself in high school calculus:

"Psst! Integrals let us 'multiply' changing numbers. We're used to "3 x 4 = 12", but what if one quantity is changing? We can't multiply changing numbers, so we integrate.

You'll hear a lot of talk about area -- area is just *one* way to visualize multiplication. The key isn't the area, it's the idea of combining quantities into a new result. We can integrate ("multiply") length and width to get plain old area, sure. But we can integrate speed and time to get distance, or length, width and height to get volume.

When we want to use regular multiplication, but can't, we bring out the big guns and integrate. Area is just a *visualization technique*, don't get too caught up in it. Now go learn calculus!"

That's my aha moment: integration is a "better multiplication" that works on things that change. Let's learn to see integrals in this light.

## Understanding Multiplication

Our understanding of multiplication changed over time:

- With integers (3 x 4), multiplication is
*repeated addition* - With real numbers (3.12 x sqrt(2)), multiplication is
*scaling* - With negative numbers (-2.3 * 4.3), multiplication is
*flipping*and scaling - With complex numbers (3 * 3i), multiplication is
*rotating*and scaling

We're evolving towards a general notion of "applying" one number to another, and the properties we apply (repeated counting, scaling, flipping or rotating) can vary. Integration is another step along this path.

## Understanding Area

Area is a nuanced topic. For today, let's see area as a *visual representation of of multiplication*:

With each count on a different axis, we can "apply them" (3 applied to 4) and get a result (12 square units). The properties of each input (length and length) were transferred to the result (square units).

Simple, right? Well, it gets tricky. Multiplication can result in "negative area" (3 x (-4) = -12), which doesn't exist.

We understand the graph is a *representation* of multiplication, and use the analogy as it serves us. If everyone were blind and we had no diagrams, we could still multiply just fine. Area is just an interpretation.

## Multiplication Piece By Piece

Now let's multiply 3 x 4.5:

What's happening? Well, 4.5 isn't a count, but we can use a "piece by piece" operation. If 3x4 = 3 + 3 + 3 + 3, then

3 x 4.5 = 3 + 3 + 3 + 3 + 3x0.5 = 3 + 3 + 3 + 3 + 1.5 = 13.5

We're taking 3 (the value) 4.5 times. That is, we combined 3 with 4 whole segments (3 x 4 = 12) and one partial segment (3 x 0.5 = 1.5).

We're so used to multiplication that we forget how well it works. We can break a number into units (whole and partial), multiply each piece, and add up the results. Notice how we dealt with a fractional part? This is the beginning of integration.

## The Problem With Numbers

Numbers don't always stay still for us to tally up. Scenarios like "You drive 30mph for 3 hours" are for convenience, not realism.

Formulas like "distance = speed * time" just mask the problem; we still need to plug in static numbers and multiply. So how do we find the distance we went when our speed is changing over time?

## Describing Change

Our first challenge is describing a changing number. We can't just say "My speed changed from 0 to 30mph". It's not specific enough: how fast is it changing? Is it smooth?

Now let's get specific: every second, I'm going twice that in mph. At 1 second, I'm going 2mph. At 2 seconds, 4mph. 3 seconds is 6mph, and so on:

Now this is a good description, detailed enough to know my speed at any moment. The formal description is "speed is a function of time", and means we can plug in any time (t) and find our speed at that moment ("2t" mph).

(This doesn't say *why* speed and time are related. I could be speeding up because of gravity, or a llama pulling me. We're just saying that as time changes, our speed does too.)

So, our multiplication of "distance = speed * time" is perhaps better written:

where speed(t) is the speed at any instant. In our case, speed(t) = 2t, so we write:

But this equation still looks weird! "t" still looks like a single instant we need to pick (such as t=3 seconds), which means speed(t) will take on a single value (6mph). That's no good.

With regular multiplication, we can take one speed and assume it holds for the entire rectangle. But a changing speed requires us to combine speed and time piece-by-piece (second-by-second). After all, each instant could be different.

This is a big perspective shift:

- Regular multiplication (rectangular): Take the amount of distance moved in one second, assume it's the same for all seconds, and "scale it up".
- Integration (piece-by-piece): See time as a series of instants, each with its own speed. Add up the distance moved on a second-by-second basis.

We see that regular multiplication is a *special case* of integration, when the quantities aren't changing.

## How large is a "piece"?

How large is a "piece" when going piece by piece? A second? A millisecond? A nanosecond?

Quick answer: Small enough where the value looks the same for the entire duration. We don't need perfect precision.

The longer answer: Concepts like limits were invented to help us do piecewise multiplication. While useful, they are a *solution to a problem* and can distract from the insight of "combining things". It bothers me that limits are introduced in the very start of calculus, before we understand the problem they were created to address (like showing someone a seatbelt before they've even seen a car). They're a useful idea, sure, but Newton seemed to understand calculus pretty well without them.

## What about the start and end?

Let's say we're looking at an interval from 3 seconds to 4 seconds.

The speed at the start (3x2 = 6mph) is different from the speed at the end (4x2 = 8mph). So what value do we use when doing "speed * time"?

The answer is that we break our pieces into small enough chunks (3.00000 to 3.00001 seconds) until the difference in speed from the start and end of the interval doesn't matter to us. Again, this is a longer discussion, but "trust me" that there's a time period which makes the difference meaningless.

On a graph, imagine each interval as a single point on the line. You can draw a straight line up to each speed, and your "area" is a collection of lines which measure the multiplication.

## Where is the "piece" and what is its value?

Separating a *piece* from its *value* was a struggle for me.

A "piece" is the interval we're considering (1 second, 1 millisecond, 1 nanosecond). The "position" is where that second, millisecond, or nanosecond interval begins. The value is our speed at that position.

For example, consider the interval 3.0 to 4.0 seconds:

- "Width" of the piece of time is 1.0 seconds
- The position (starting time) is 3.0
- The value (speed(t)) is speed(3.0) = 6.0mph

Again, calculus lets us shrink down the interval until we can't tell the difference in speed from the beginning and end of the interval. Keep your eye on the bigger picture: we are multiplying a collection of pieces.

## Understanding Integral Notation

We have a decent idea of "piecewise multiplication" but can't really express it. "Distance = speed(t) * t" still looks like a regular equation, where t and speed(t) take on a single value.

In calculus, we write the relationship like this:

The integral sign (s-shaped curve) means we're multiplying things piece-by-piece and adding them together.

*dt*represents the particular "piece" of time we're considering. This is called "delta t", and is not "d times t".*t*represents the position of dt (if dt is the span from 3.0-4.0, t is 3.0).speed(t) represents the value we're multiplying by (speed(3.0) = 6.0))

I have a few gripes with this notation:

- The way the letters are used is confusing. "dt" looks like "d times t" in contrast with every equation you've seen previously.
- We write speed(t) * dt, instead of speed(t_dt) * dt. The latter makes it clear we are examining "t" at our particular piece "dt", and not some global "t"
- You'll often see , with an
*implicit*dt. This makes it easy to forget we're doing a piece-by-piece multiplication of*two*elements.

It's too late to change how integrals are written. Just remember the higher-level concept of 'multiplying' something that changes.

## Reading In Your Head

When I see

I think "Distance equals speed times time" (reading the left-hand side first) or "combine speed and time to get distance" (reading the right-hand side first).

I mentally translate "speed(t)" into speed and "dt" into time and it becomes a multiplication, remembering that speed is allowed to change. Abstracting integration like this helps me focus on what's happening ("We're combining speed and time to get distance!") instead of the details of the operation.

## Bonus: Follow-up Ideas

Integrals are a deep idea, just like multiplication. You might have some follow-up questions based on this analogy:

- If integrals multiply changing quantities, is there something to divide them? (Yes -- derivatives)
- And do integrals (multiplication) and derivatives (division) cancel? (Yes, with some caveats).
- Can we re-arrange equations from "distance = speed * time" to "speed = distance/time"? (Yes.)
- Can we combine several things that change? (Yes -- it's called multiple integration)
- Does the order we combine several things matter? (Usually not)

Once you see integrals as "better multiplication", you're on the lookout for concepts like "better division", "repeated integration" and so on. Sticking with "area under the curve" makes these topics seem disconnected. (To the math nerds, seeing "area under the curve" and "slope" as inverses asks a lot of a student).

## Reading integrals

Integrals have many uses. One is to explain that two things are "multiplied" together to produce a result.

Here's how to express the area of a circle:

We'd love to take the area of a circle with multiplication. But we can't -- the height changes as we go along. If we "unroll" the circle, we can see the area contributed by each portion of radius is "radius * circumference". We can write this relationship using the integral above. (See the introduction to calculus for more details).

And here's the integral expressing the idea "mass = density * volume":

What's it saying? Rho: rho is the density function -- telling us how dense a material is at a certain position, r. dv is the bit of volume we're looking at. So we multiply a little piece of volume (dv) by the density at that position rho(r) and add them all up to get mass.

We'd love to multiply density and volume, but if density changes, we need to integrate. The subscript V means is a shortcut for "volume integral", which is really a triple integral for length, width, and height! The integral involves four "multiplications": 3 to find volume, and another to multiply by density.

We might not solve these equations, but we can understand what they're expressing.

## Onward an upward

Today's goal isn't to rigorously understand calculus. It's to expand our mental model, and realize there's another way to combine things: we can add, subtract, multiply, divide... and integrate.

See integrals as a better way to multiply: calculus will become easier, and you'll anticipate concepts like multiple integrals and the derivative. Happy math.

## Other Posts In This Series

- A Gentle Introduction To Learning Calculus
- Understanding Calculus With A Bank Account Metaphor
- Prehistoric Calculus: Discovering Pi
- A Calculus Analogy: Integrals as Multiplication
- Calculus: Building Intuition for the Derivative
- How To Understand Derivatives: The Product, Power & Chain Rules
- How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms
- An Intuitive Introduction To Limits
- Why Do We Need Limits and Infinitesimals?
- Learning Calculus: Overcoming Our Artificial Need for Precision
- A Friendly Chat About Whether 0.999... = 1
- Analogy: The Calculus Camera
- Abstraction Practice: Calculus Graphs

## Leave a Reply

134 Comments on "A Calculus Analogy: Integrals as Multiplication"

Awesome! As always ;)

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@ Yifeng: Thanks! :)

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.

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.

Hi J.H., no problem. Let’s say our equation for speed is “speed at some time period is twice that time period”. For example, if we’re at t=1 second, we’re going 2 mph. At t=2 seconds, we’re going 4 mph, and so on.

We could try to plug this into the equation “distance = speed * time” to figure out how far we’ve gone after some number of seconds. We might get something like:

distance = 2t * t

However, t looks like a static quantity here. If we pick t=3 seconds, we might try to plug it in and get:

distance = 2(3) * 3

but this is saying we were traveling 6mph for 3 seconds. That’s not true, we were only going 6mph in the final second. In the beginning (at t=0, t=1, t=2…) we were going at a slower rate. So “distance = 2t * t” isn’t a good representation of our scenario.

“The answer is that we break our pieces into small enough chunks (3.00000 to 3.00001 seconds) until the difference in speed from the start and end of the interval doesn’t matter to us.” so, by increasing number of “chunks”, if we “zoom in” on one of those chunks, we are making something that looks similar to the first picture (“Understanding Area”) and area of that chunk will relatively calculate “instant” distance between some infinitesimally small time interval (3.00000 to 3.00001) where speed is ~constant? Understood all of that you said about multiplication, but when it comes to understanding multiplication with “changing speed” I can’t think a better model than the idea that 1 small chunk = 1 area that can’t be broken into more smaller chunks and that is the point where regular multiplication holds. Am I doing it right? :D Thanks!

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

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

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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.

later

T.

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

-Kalid

@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!

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. :)

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. :)

@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 :).

I always had a problem with Integral Calculus in my high school and was happy that I don’t need to study integrals anymore. but your explanation was awesome. I now really know what integration means! Thanks.

@Deepan: You’re welcome, happy it helped!

Congrats on a great explanation. I think I have always been able to envision integrals properly–because I could always see when it’s to be applied–but I never figured out that it was generalized multiplication that I was seeing. Of course, I am kicking myself at how seemingly obvious the integral’s identity is. :)

@CJ: Great point. Another way to see the integral is multiplying the average value of a function over the interval you’re considering.

…presumably that’s if you are interested in the final result of the integral and not the process via which it is obtained…

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A most enjoyable read. This is probably your best article yet!

I took Calculus BC back in high school, but I was forced through all the material at breakneck speed. I scored great on the test but was left feeling it was all a waste of time.

Now, thanks to you and other sites and books, I’m combing through the material again in an attempt to truly understand it. Incredibly- it’s actually fun! I hope to pass my enjoyment along as a tutor someday.

Thanks for the work you’ve put into these articles. Your belief seems to be that anyone can understand something if it’s only explained properly. If I’m correct in that, it’s a fantastic attitude to hold.