A Calculus Analogy: Integrals as Multiplication

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. 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:

\displaystyle{distance = speed(t) \cdot t}

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

\displaystyle{distance = 2t \cdot t}

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:

\displaystyle{distance = \int speed(t) \ dt}

  • 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 \displaystyle{\inline \int speed(t)}, 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

\displaystyle{distance = \int speed(t) \ dt}

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:

\displaystyle{Area = \int Circumference(r) \cdot dr = \int 2 \pi r \cdot dr = \pi \cdot r^2}

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":

\displaystyle{mass = \int_V \rho(\vec{r})dv}

What's it saying? Rho: \displaystyle{\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 \displaystyle{\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

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

Questions & Contributions


  1. 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.

  2. @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.

  3. 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.

  4. @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.

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

  6. 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. :)

  7. 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. :)

  8. @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 :).

  9. 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.

  10. 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. :)

  11. @Nobody: Thanks, glad you liked it! I’m the same way, I used integrals for a long time without really getting what they were about. They seemed so disconnected, or like a special operation.

  12. Well written! Another angle on understanding integration is to consider the average value of a function. Then the area of a circle becomes the average radius times the circumference, or alternatively the average circumference (which is pi * r) times the radius.

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

  14. 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.

  15. @Mr. ag: Thank you, glad you liked it! I’m definitely a believer that any subject can be understood if explained properly. Our ‘basic’ reading, writing, arithmetic and algebra skills were once considered extreme specialties a few hundred years ago. Our brains are capable of so much if subjects are presented well :). Appreciate the comment!

  16. Great explanation.i’ve really enjoyed your articles.could you please explain how the formulas for integration came into existence(like integral of cosx=sinx).i was eagerly looking forward your next article in this topic.

  17. finally after all these years i am beginning to make sense of the purpose of integration in real life. well not really. its just as confusing 😉 but your article is definitely helping me see calculus as something beyond 50 marks worth of questions that it was in the 12th standard.
    you are gifted at explaining things. its truly a gift to be able to to break things down to simple levels and explain it to people. as a novice teacher i know how tough it is to teach!! keep at it !!!

  18. This is actually how my math professor explained it. He gave us the theory after explaining it in this way. I also got a double dose as my physics professor explained it the same way and I took physics and calculus as co-reqs.

    As a side effect, I always wondered why others said calculus was so hard. Now I guess I understand where they are coming from.

  19. I’m surprised you used ‘d’ for Delta instead of the standard Greek letter `delta’.

  20. @vishwanath: Thanks for the encouragement! Sometimes I can only reach insights after banging my head against the wall, I’m really happy you found it useful!

    @sil-chan: That’s awesome, I wish more teachers taught it that way! Only showing the formal book definition tends to confuse students.

    @William, Nobody: Yeah, I didn’t want to use the actual Greek letter — too hard to type in plain HTML :).

  21. What the… Integration as analogous to multiplication?? I never thought of that. How’d you ever come to that “aha!” moment

  22. @love-hate: Glad it helped! I intuitively see integration as a bunch of little additions, and multiplication is like a bunch of additions also. Also, the units end up being the same (integral x dx has units x squared, x times x has units x squared).

  23. hmmm I’m having trouble seeing bow we can choose small enough pieces so that all the values look the same…

    For example, if y=x don’t the values change no matter how small the piece you choose?

  24. Great example of analogy! I really like it!

    Recently I started posting interestnig analogies I found on the web on blog.ygolana.com. I thought it could be a good idea to create a place where people can help each other to find useful analogies so I created a simple site (www.ygolana.com) Check it out!

  25. @Frank:
    It might make more sense if you imagine dividing up the area between the x-axis and the function y=x into many vertical rectangles, and adding up their areas. The more rectangles you use, the better the approximation of the area. The idea behind integration is that if I divide up the area into infinitely many rectangles with infinitely small width, no matter how far you “zoom in”, you’ll never see the difference between the “real” shape (which is triangular) and my “approximated” shape (which is composed of many rectangles). So it’s reasonable to say that the area is in fact the same. Now how exactly do we add up infinitely many infinitely small things to get a real number? Uh…Kalid?!? 😉

  26. @Peter: Cool, I’ll check it out!

    @Matt: Thanks for the comment! One of the hardest parts is getting my head around the idea of “accurate enough”.

    Here’s how I think about it. In real life, we hit this all the time: A screen image is a grid of pixels, yet we can see perfectly smooth shapes like curves, circles, faces, etc. Similarly, inkjet printers spray a matrix of dots on a paper, but to us it looks like a smooth unbroken image or line.

    The key is realizing that the approximation is only an approximation at that higher level of accuracy — at the level that we work at, it appears indistinguishable from the real thing. Calculus helps formalize some of these ideas with limits (informally, two numbers that have a difference less than our error margin appear the same to us).

    Unfortunately, we don’t really talk about this much, and we sometimes say numbers are equal, and sometimes say they aren’t. There’s a notion of infinitely small numbers which makes this clearer, and is used in physics. That is, you can talk about how infinitely small numbers interact with each other, and with infinity, to give numbers we can detect. A poor analogy but it may work: A caveman could probably not conceive of an individual atom, or the gargantuan Avogadro’s number (6 x 10^23), but when this tiny particle and huge number combine we can get something we can detect.

    The key is writing this idea down in the language of math: numbers that are too small and too large for us to detect can interact to give us numbers we can work with.

  27. Hi Kalid,

    your explanations of the underlying concepts of mathematics do bring the subject at a democratic lavel, a level on which people communicate, collaborate and work towards making the subject useful for greater number of people.

    Now coming to the subject, cud i say dat, differentiation is inverse of integration?
    and going by dat if i have to apply differentiation, let’s say on the example of circle, all i have to do is to run a playback, i.e. to peel of all those tiny rings (or, in other words) thus divide the circle into the tiniest possible rings.
    Once i m done peeling… I would measure this ring, to see the result of differentiation application, which should be 2*pi*r.
    BTW, would not I b applying multiplication again, to measure that tiniest ring, i.e. finding the area of that ring, pi*r^2?

  28. One way I understood the basic integral notation is with my crude understanding of sets and functional programming. Using the given example above (speed and time):

    There’s an implied set of values of time, and we take a piece of it or a member of that set. That becomes the slice of time. We then apply it to a function of time that is speed. This results in another set whose members are results from each function result using the said function given a slice (or element of implied set of time) as input. Finally, we apply the ‘integrate’ operator, or probably a ‘map’ to the ‘integrate’ function; or, to put it simply, use the integrate function on all members of the resulting set to return the integrated value.

    or something like:
    map(‘integrate’, (getSpeed(t) | t <= time_slices))

  29. note to extend idea by Kalid above:

    circumference of a cirle (a 1d distance) = 2 pi r
    area of a circle (a 2d area) = pi r ^ 2

    (note the integral /derivative of each other.)

    surface area of a sphere = 4 pi r ^ 2
    volume of a sphere = 4/3 pi r ^3

    try this with squares and cubes… hint, base it on the shortest distance from the centre to a side.

    how cool is that!

  30. I have understood the basic idea of calculus. Now i am getting problems in solving word problems on differential equations, like growth and decay problems. The logic behind variable separation etc. Can you guide ? If you respond i can write in more detail.

  31. Currently studying calculus. This is a fantastic explanation! Never seen anything this lucid and clear – it has really helped. Thank you!

  32. @apoorva: I’d love to do more on differential equations, but don’t have the necessary background yet. Once I learn more, I would love to write about them.

  33. (Disclaimer: ignore me as much as desired, it’s midnight over here, and besides I have half a high school year of experience with calculus and we won’t start integrals until next year.)

    Arithmetic multiplication is a special case of integration, yes? Then what is this special case? Integration of a constant function?

    Mmm… That’s an interesting way to realize that, if c is constant, the equation
    integral (c) dx = cx
    simplifies to
    c*x = cx

    We would like to imagine that on some higher plane of existence, pandimensional beings learn calculus first, and then read esoteric texts on quaint ideas of ‘multiplication’ and ‘division’ that lesser mortals are more comfortable with.

    Oh, oh, and what about arithmetic division? Differentiating a constant function gives you zero, so that won’t work. What you could do is take the dividend as a product cx, as in c times the divisor. dcx/dx = c. So, apparently, if you want to use a special case of differentiation for arithmetic division, you would have to find the quotient beforehand. This is hilarious, and I like to think I’m giving some pedant an apoplectic fit somewhere.

    Great site, Kalid! It may be difficult to see what kind of intuitive muddle your articles are inspiring in me, but it’s clearly something, and something’s better than nothing! Or something.

  34. Oops, I didn’t realize the first two equations contract into each other. integral of c over x equals product of c into x. You don’t need to equate two equations, one of which is gratuitous and obvious.

    Now if only teachers could point out that particular formula (integration of a constant) as the Special Integral of Multiplication; my interest would multiply similarly.

  35. @Offendi: Awesome comment, thanks for the thoughts. Yep, somewhere out there, calculus could indeed be the starting block :). Division is interesting — derivatives aren’t *quite* division, they’re more the “application of division on a point-by-point basis” — that is, figure out a ratio for how much output I get for a certain amount of input. It’s more complex, and need to think more about how this reduces to normal division.

    Thanks for the warm words, always happy to muddle or unmuddle where I can.

  36. Division itself (outside of the integer/integer case) is a very odd operation. I will give an explanation of what is being done, and the derivative will hopefully be illuminated.

    Let us divide -pi/e. This has so many oddities from the integer/integer case: the magnitudes are irrational (and the quotient will be irrational), the numerator has a negative sign, and the magnitudes are transcendental (that means no polynomial division). How are we going to solve this? Let us look for useful properties of one that we can solve intuitively:

    4/2 is the division of 4 objects into 2 groups. There remain 2 objects per group. So, 4/2 = 2 or 4 = 2*2. Now, let’s generalize the denominator, 4/x = m or 4 = m*x. Finally, let y = 4 and we get y = m*x. This reminds you of the equation for a line through the origin in the Cartesian plane! Returning to -pi/e, let us plot (e,-pi), draw a line between that and the origin, and solve for the slope of the line. (That is why division by zero should be left undefined.)

    Now, let’s find the derivative of f:R->R defined by f(x) = x^2 + x at x = 4. So, we plot y = f(x) and draw the line tangent to f at x=4. We notice immediately that the y-intercept is not zero, it is -16. So, let’s “normalize” this by shifting up sixteen units: y = x^2 + x + 16 is what we now plot. You will notice that the derivative at x=4 is simply y(4)/4.

  37. Note that I came up with that just now, so I leave it to you fill in the details. For instance, I realized that normalization in the sense of a shift is unnecessary, but it does well illustrate the connection (when x is nonzero).

  38. Fantastic website. Exactly what I needed to brush up my maths. I can’t wait for a part on differential equations as I would love to understand quantum mechanics.

  39. Hello!

    I’ve always been a little math-phobic. Since I arrived at college though, I concluded that I wanted to study Economics, which eventually requires a lot of math courses, so I have been trying to relinquish myself of my math-phobia. My intuitive grasp has always been my strongest muscle, so I was overjoyed to find your site, replete with information on calc! And you have an ebook!
    Consider me another new reader and customer.

  40. @Chris: Awesome! I really think there’s a need to have a site focusing on the intuitive insights that really help things click, vs. just listing out the details that are available everywhere :). Appreciate the support!

  41. Wow, nice explanation. I’m an engineer who’s been doing this for a few years and haven’t understood it for years, until now. Uni would’ve been so much more enlightening had I seen integrals that way.

  42. Even though this approach “makes sense” when looking at definite integrals (which by definition are Riemann sums), the mental model of “multiplication” has almost nothing to do with the process of integrating (or, rather, indefinite integration) which is a purely algebraic (as opposed to arithmetic) process.

  43. @Anonymous: Yes, indefinite integrals are an interesting beast. In fact, there’s not really a “process” for them, it’s more “What function has the derivative that we want?” I.e., we aren’t finding integrals directly, we know them because such and such a function has the derivative we’re looking for. This confused me a for a long, long time as finding derivatives is so mechanical, and finding integrals so free-flowing.

  44. Hi,
    Thank you for this post!

    Regarding the first example with the formula Distance = 2*t *t, in my opinion the accompanying drawing is not correct. There should have been a set of rectangles, not triangles, otherwise the formula should have been Distance = (2*t*t)/ 2.

    Regards and thanks again

  45. @thereader: Thanks for the comment. I put in the triangle (it could have been a curve) because I wanted to show the actual speed over time, vs. the “computed” formula we’d get with the rectangles. distance = 2*t * t is actually meant to be incorrect, to show we can’t just plug in speed * time directly.

  46. Dear Kalid

    I have really enjoyed my “a-ha” moment after reading this article, one thing I would like to ask that how can I intuitively explain the multiplication of exponential with cosine, and also how can I integrate both of them.

    Waiting for your reply


  47. you know I always had a problem with the little triangle immediately under the curve and how we computed it’s area…no….it was more of an obsession, and that was a REAL barrier to my comprehension of integral calculus.
    If I understand what you are presenting here my worries are over, I can sort of ignore it because I am just computing the area of each rectangle and adding them up…….am I right?

  48. @jewill: Awesome question — that’s the heart of Calculus, “what can you ignore?”. Here’s how I see it: the little triangles (immediately under the curve) are measurement errors due to the instrument itself (the particular width of the rectangles — smaller rectangles have smaller triangles).

    Imagine a doctor measuring a patient’s heart rate with a machine. The very presence of the machine makes the patient nervous, so his heart rate is different from the “real” value (if the machine weren’t there). In a similar way, taking specific-width rectangles creates a slight difference in area compared to the perfect function.

    Most functions (basically, continuous ones) are well-behaved so the difference can be ignored. We can say “If our rectangles were perfect, what could we measure?” sort of like “If our machine was so small the patient didn’t notice it, what would we measure?”. Many functions behave as we expect under this prediction (some other functions don’t, so we can’t use calculus on them).

  49. I’m not sure if it’s an american VS british thing, but i’m pretty sure i was taught calculus using infentesimals before i was taught it with limits. (well, kinda simplified, not so rigorous, ‘it kinda works but we’ll do it rigorously after you get the basics’ way)

    so the way i learnt it, ‘d’ is an infentesimal difference (as opposed to capital delta, which is a larger difference), if you do d*x and d*y, the ratio of dx/dy and x/y stays the same, but dx/dy is basically instantaneous and when you’re dealing with straight lines it doesn’t matter. so you can use deltax/deltay to find the gradient instead of needing dx/dy which basically shrinks the difference to be really tiny.

    similarly, when integrating, you’re multiplying every instant of your f(x) with dx which is so small it’s basically a point, and adding them together. take an infentesimal distance, and multiply it by x to scale it up so it still works. i can’t really explain this but this is how i first understood it (while always having at the back of my mind, ‘it doesn’t QUITE work like this, but it’s close enough for now).

  50. @rash: Thanks for the note. Yep, unfortunately the subtleties of infinitesimals are usually lost in class. In mine, we did limits for a few weeks then “introduced” the idea of the derivative [which seems backwards — limits were only invented to patch holes in the rigor of derivatives, but now are taught first!].

  51. gr8 way to learn the concepts. thank you so much Kalid. Now I understand , it’s criminal to teach calculus without physics.

  52. I loved this series. Math can be so very interesting when you talk about what it does, as opposed to how to do it. I find Math as Ideas more useful than most Math as Technique. I enjoyed all math when I was in school, all the way to Advance Engineering Math, but this is more interesting.

    I’m going to reread this several times, it makes for a great read.

  53. Hi Lance, really glad you enjoyed it! I completely agree, I want to understand the central idea first, then the mechanics of the techniques.

  54. As-salaamu Alaykum…thanks a ton!!! I’d never been able to understand integration…around 3-4 people tried teaching me integration, none were successful! Thanks to you, I now understand integration pretty well cuz you taught the basics! 😀
    I was just like “AH-HA AH-HA AH-HA” all the time!!! JazakAllahu Khayran Katheeran!!! 😀

  55. Thanks for the help. A good instructor is worth their weight in gold. I do not have a math brain and cant understand the language. Is there a drug that I can take to allow me to at least understand Calculus as it is written even though I cant use it. ??

  56. Finding the area is truly a graphical representation of what is actually going on, as a tool to help understand. For example, with a graph of Force vs. Time, the act of integrating – or finding the area under the curve of the function (which means the force varies) – is actually simply getting Force x Distance = Work. Integration is just multiplying variable factors, which we cannot do with usual math.

  57. Your post is wonderful! I will definitely use this as a reference when I teach my class integrals. One small thing though– Fubini and Tonelli would cringe when you say

    “Does the order we combine several things matter? (No)”

    Obviously that doesn’t matter at this level, but I just had to point it out :)

    Once again, great job!

  58. @Raymond: Yep, exactly. Finding area is one visualization. But we need to make sure not to get stuck there, and see integration as a more generic “combination” of two changing variables. Otherwise we think it’s only useful for solving explicitly geometric problems.

  59. @G: Great point! I didn’t realize that the order of integration could matter (at the level I studied Calculus, it never became an issue). Really neat though, thanks for the clarification!

  60. Great!!! Just what I was looking for! Writing simple explanations for something
    more abstract is highly underrated, although it is so much more useful than
    re-phrasing abstract concepts, which incidently doesn’t mean one has really understood it 😛

  61. Great article, as all of yours are. But can you please explain why “3.12 x sqrt(2)” is “scaling”? Thank you!!

  62. This is fantastic, but the whole analogy only seems to go so far, since single integration isn’t quite communicative. Integrating force with respect to distance (integral of F(x)dx ) is much different than integrating distance with respect to force ( integral of x(F)dF). Why is this though? I mean, if the integral is truly an advanced form of multiplication shouldn’t it have the properties of multiplication as well?

  63. Hi David, great question. I’m not well-versed in mathematical analysis (never taken it formally! one day…), but here’s my understanding.

    Physics can get confusing because we usually have a well-known independent variable (time, distance, etc.) and integrate with respect to that. It’s really weird to invert the equation and have “time” be a function of some other process [time flows independently, it isn’t flowing *because* we are moving!]. But in math, y = f(x) can easily be x = g(y) [where f and g are inverses], and you can certainly integrate with respect to one variable or another. In physics, we’re always finding f(t), not rewriting “t” in terms of some result.

    But from a math perspective, we could: the bounds of integration would change, but there’s no mathematical reason we can’t integrate either f(x) or g(y). Imagine you’re finding the area of a circle: you find the top semicircle by integrating y = sqrt(1 – x^2) from x=-1 to 1. Or, you could integrate x = sqrt(1 – y^2) from y=-1 to 1 [the right semicircle]. From a math perspective, you just changed variables. In physics, we know one variable is “supposed” to be independent and it hurts our brains :).

  64. In response to David, I think one needs to differentiate between path and point functions (or scalar and vector functions). The example used of Integrating force with respect to distance (integral of F(x)dx ) is a path function (Work), if a comparable analogy to multiplication is to be made, then one needs to think along the lines of vector multiplication or (cross product) in which case AxB != BxA. I believe using vector calculus a similar analogy to the one here can be made but it may not be easily understood.

  65. In the Describing Change section, you say “Take the amount of time moved in one second,” when seems a little off. The time elapsed in one second is a second. Don’t you mean the distance moved in one second or the speed after one second?

  66. That’s really really awesome. I had a faint idea about this multiplication as i looked at how units are conserved in integration. But this shed a whole new light onto it. PLEASE write more about integration…

  67. A big big thanks from the bottom of my heart……they used this concept in school and I was like dafuq is happening! Thanks a lot…all I needed was just a perspective to look at integration…Thanks again :)

  68. Hi, thanks for this post. There’s just one detail I can’t seem to understand. I can’t seem to understand what values does “dt” take in this “improved” multiplication.

    Namely in: distance = {integration sign} speed(t) . dt

    If dt is a very small piece (infinitesimally small) shouldn’t its “size” be very close to 0? In fact, why isn’t its size 0, if it’s “infinitesimally” small?

    For example, if dt is shrinked down to 3.0000001 – 3.0000000 = 0.0000001
    In my head it makes sence, : )
    I can actually multiply this piece with its value at speed(t), and get one item of the lot of things I’m adding up.
    But if dt is infinitesimally small (and not 0.0000001, like in my example) what size of it am I actually multiplying to its value in speed(t)?

    Do I make any sense at all?

    In my head, from what you explained in this post, integration comes down to adding up an infinite Set of numbers (which is a Sequence).

    So is it possible to express : distance = {integration sign} speed(t) . dt into Sequence notation? Where dt would have a size, which is multiplied by its value in the function speed(t), to give the items/numbers of that Sequence?

    Thanks for any answer, I hope I didn’t make this overly confusing…

  69. Hi Stephen, great question. The role of “dt” (or dx, etc.) is one that bothered mathematicians for many decades, and only got resolved 100 years after Newton died.

    Here’s a quick analogy: how big is a point on the number line? Let’s say we draw a point at x = 3, how big should we make the dot?

    If we give it a width, then there’s no room for the nearest neighbor, i.e. the point at x=3 overlaps with x=3.0000000001.

    Ok, fine: let’s say the point has zero width. Well… then it’s not there, is it? What’s the difference between a “zero-width” point and complete empty space? How do you know the difference?

    The key insight in calculus was to create a theory like this:

    Let’s work out the math *as if* a variable could be split into points of a set thickness (t gets broken into points of width dt, x into dx, and so on). This gives us a model for behavior, and we might make predictions with this knowingly-flawed model. For example, the “dx has size” model might predict that x^2 changes by 2x + dx.

    Now here’s the big trick: what does the model say will happen if the points were perfect? (I.e., there was no gap between them?) If the points were perfect, then dx = 0, and the model says “x^2 changes by 2x”.

    And that appears to be the case! So, if you like, dx (or dt) is an artificial measurement we introduce in order to get a model, then we make it perfect by pretending dx was never there.

    An analogy: suppose you measure the heart rate of patients in a hospital. Well, patients are more nervous in the hospital than at home, so you account for that effect. You need the hospital to make the measurement, but once you have the measurement, you pretend the hospital environment was never there and adjust the reading.

    “dt” is the little segment that we pretend we can split t into in order to get the reading (dt has the same units as t, so if t=seconds, dt is still seconds, but likely .00000000001 seconds; the exact number isn’t specified, it’s just “small”).

    This article has a bit more which might help:


  70. Thanks a lot Kalid, especially for the quick reply, and it already makes total sense. This issue had bothered me since secondary school actually… I just couldn’t picture the use of the dx bit in equations. But I can see how it’s useful now with your analogies.
    I’ll check the Limit post ASAP for an even better understanding of the topic.

    Thanks again!

  71. Awesome! More than happy to help. Getting an intuitive understanding for what dx, dt really meant is something that I’ve struggled with since school as well. For me, the “heart rate” analogy works, i.e. we need an instrument of some kind to make a measurement, but we need to remember that the instrument itself can affect the outcome, and adjust for it.

    Another way to put it: “The instrument says the patient’s heart rate is 90 beats/minute + the weight of the instrument.”

    There’s a weird paradox where we ask the instrument to predict what would happen if it weren’t there.

  72. great explaination ! But integration assumes that the quantity is changing continuously.the time interval we are taking is infinitely small. But what if it has a finite length?if in real life quantities like speed do not change continuously but in the form of jumps then it wont be possible to add up the small distances.i mean if there is a minimum finite change possible in changing physical quantities such that a smaller change isnt possible then integration wont be accurate in such cases.but it is the most convenient possible method we can use.your explaination was really really great.

  73. hiii…thanks a lot for the explanations…now calculus is more fun rather than a nightmare…but can u please explain how we can simplify problems by understanding…i meant can we extend this idea into difficult problem solving???

  74. your’e the sort of teacher i’ve always dreamed of ………someone who just emphasis on the most basic things rather than just doing problems without understanding the core .
    this has always restricted students like me to improvise and to improve …
    THANK YOU for your noble gesture towards hundreds of math lovers

  75. I’m posting this so that others can read just incase they have the same question as me.

    I’m not completely sure what you mean in this section: [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.] Could you explain it a little more? Thanks!

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