Riemann Integrals





It’s not at all difficult. 
What is it you want to know? You want me to explain it to you in a kind of formal logic fashion?

Hmm… I suppose I could. It’s easier to do if you have visual aids.

Alright, let me give a general summation: We all know that lines are named by where they go, yes? The road from Glasgow to Edinburgh. Point A to point B. And if you have many roads, you may have to be more specific: Route Interstate 5 from San Diego to Washington State. 

Let’s say that a simple, straight line could be named by two points. Line AB or Line TJ. You may have seen this in Geometry. But this isn’t terribly descriptive. No not at all. Interstate 5 may go from San Diego to Washington, but it meanders around quite a bit. It’s very wiggly.

So instead of just naming a line AB, we name it more descriptively, with a “function”. You’ve seen f(x)? This just means there’s a car on the road tracking the wiggles and creating a mathematical sentence to describe the path of the road.

A simple function or f(x) would be f(x) = 2x – 6, Which means, for every point you have…every “x” you will modify it in those ways to get the second coordinate in space. If we put in an 8 for x, we get 10. For x=8, the second coordinate or “y/ f(x)” is 10. “8,10″ And so by putting in a series of numbers we come out with a series of coordinates that create a rather good description of our line!

We name the road by how we are tracking its progress. The function, f(x) is the name of the line.

Why am I demonstrating such basic algebra, well…because sometimes people get so confused by nomenclature, that it all becomes a jumble of letters and numbers and it is confusing. Just always remember that f(x) is the description of the line, its name.

Now what if you want to know the total area in miles that lies, oh lets say, on the western side of Interstate 5, from the road to the coastline. That’s an odd shape, right (pretend the coast is a flat line, for the purpose of the visual)? You want to know how much land lies to one side of the road, but with such an irregular shape, how are you going to figure out that area?

It’s simple! You’re going to create a series of rectangles beneath that line and add them up!

Rectangles are easy, right? Their area is just height multiplied by width! A rectangle 6 feet ling by 2 feet wide has an area of…12 feet! Now do this hundreds of times!, You will have a truly good approximation of the total area to one side of that road. Let me give you a visual!


So in the above image, lets say that the road is called f(x). The line at the bottom (the x axis) is the California Coast, and the Line at the left (y axis) is the state line between Mexico and California. F(x) goes clear from Mexico to Canada, but we only want to know the land measurements from San Diego to Seattle, or…A to B.

The line A and the line B are our bounds. The top boundary is the road, the bottom is the coastline, and we want to calculate all that is between, so we are going to make rectangles, BUT! The thinner our rectangles, the more accurate our assessment will be, because there will be less left over space.


So how do we write this down?


So, f(x) is just our road, our line. That long “S” shape there on the left is just saying “This is the integral” (or the space beneath the line), and the little a and b simply indicate between boundary A and boundary B! “dx”, stands for “delta” or “change” along the x axis…or…just means “little pieces of x”, like mile markers or segments between cities.

Now obviously, this formula assumes you don’t know anything about anything and are making generalizations, like saying “all cars have four wheels”. Yes, thank you, but you’re not giving me specifics. I need specifics. So instead of just “insert line name here” we will insert the actual name of the line, the line’s function. And instead of the A and B, we will put in our actual boundaries.


Yes, i borrowed this equation because I couldn’t think of one off the top of my head. Sue me. It’s been about 30 years since I last did this nonsense.

So…On the line (3x^2 + 4x + 1) we want the area from boundary 1 to boundary 5 on the x axis, for all the “dx’s” along the x axis between 1 and 5.

Now like there are in all things, there are rules for how we do maths. There’s the rules about the order in which we do things (order of operations) and so forth. And this is no exception. There’s the Integral Rule of Powers, which tells you how to get from the above equation to the next stage (In case you’re wondering, the integral rule of powers is that the exponent gets one added, while the integer is divided by that sum. 3x^2 becomes x^3. ), but I’m not teaching you how to calculate, I’m just giving you a visual idea of how this works.


“Simon! Stop! What the f(x) are you doing!”

Calm down. The integral rule of powers simply converts our equation into the business on the top right there. Then we put in 5 for x and then 1 for x, and subtract. 


We made the line. We drew in all the little rectangles from mile marker 0 to mile marker 5, then we subtracted all the rectangles between 0 and 1, because we wanted to start from 1. In my first analogy Zero = Mexico, 1 = San Diego, and 5 = Seattle. We aren’t finding the area from Mexico to Seattle, but from San Diego to Seattle, so we just subtract all that business between Mexico and San Diego.

Now the line determines the upper edge of the rectangles. Said another way, the name of the line becomes the dimension by which we calculate how tall our rectangles will be, and so the line must be used as one of the length/height requirements. So our rectangle width is “Some distance along x” and our height increment is “However tall that line is at that point”.

So really, all you’re doing is fancy addition. Adding easily calculated shapes together and averaging a bit, to get the area of an odd, curvy shape.

Now…things get complicated if we want to find volume, but you didnt ask me that.

But what does all of this mean?

Well, friends, a line isn’t just a line. A road, isn’t just a road. What if the road was through space and time? What if the “area under it” was actually something incredibly complicated.

It’s a poem. An integral is a way of approximating a truly complicated reality in a way that makes it visually accessible. That’s all. 

Nothing complicated.

If that doesn’t clarify…I am sorry to say I am not a magician. Just a humble monster.

I’m gonna tell all y’all, it’s good to understand how we calculate integrations and what it means, but just about every usage of this calculation in the working world will come in the form of a computer automatically graphing the curve(s) and integrating for you, you just set the rules and boundaries and exclusions. So don’t freak out too much over it, just understand it. -signed, a chemist who literally uses a program to do exactly this every single day.

Quite. Because it’s calculation.

I’m always saying that you have been taught nothing but calculation, because at some point, long ago, men decided that they’d rather grasp the busy work, and that this is the only way their brains could come to comprehend the true complexity of abstractions…until they got so bogged down in the minutia they forgot they were on a quest to comprehend the universe.

I taught you this so that this tiny calculation makes sense as a concept, not so that you actually ever have to do it. As Duck says, you made calculators so they would do it. 

The flesh on the bottom of your feet is capable of becoming rather thick and coarse, to protect you as you walk around. But then you invented shoes, so that you could move over all terrains, and you never thought once about going back. Now everyone who can, wears shoes, because of course you would, and no one ever says “But if you don’t go barefoot, you won’t understand having feet!” because that is stupid.

I say, “Oh look, we can calculate the area beneath a cruve, or the volume or the sound levels, or whatever, using definite and indefinite integrals, and here is the formula” and you say…

“Great! Let’s build a machine to do that with accuracy every time!”

And I say… “Yes please, let’s because it’s faster and humans are rubbish at math.”

Yep, working as an engineer building a mathematical model of some of our tools involving fluid mechanics and other stuff that tends to make peoples’ eyes glaze over if it’s not explained well…I COULD calculate out differential equations, but it’s faster and more accurate if I just tell Matlab to do it. If it wasn’t for the fact that I’m the only one on my team who still knows how fluid mechanics work, I would be able to say that 90% of the calculations I need to do are already done by Excel spreadsheets created years ago where all you have to do is input your values. As it is, I’m having a hell of a time getting somebody to review my work because no one other than the PhD has a better grasp of it than I do, and I’m starting to wonder about him. I keep hearing murmurings about turning it into software somewhere down the road for purposes of design optimization which currently has to be done by, uh, well, me.

Anyways, that went off on a bit of a tangent, but my point is that even as an engineer, knowing how to do the calculations doesn’t have much value. Understanding the theory does. If you know how something works, you can arrive at the calculations yourself.

I know several professional mathematicians and every single one of them routinely says, whenever I ask a math question, while I am inventing or thinking about scientific things, “I’ll have to refresh myself, because I haven’t done any of that math in about x years.” or “Hang on, let me look up the formula again, so I can check that.”

The fact that high school teachers demand you calculate endlessly and see the ability to calculate as an accurate check to the grasping of an abstract concept is both preposterous and hilarious to me. 

We haven’t even addressed the benefit of “indefinite integrals” and the ability to determine incredible information simply by seeing the x and y axises as things other than increments of heigh and length. When they turn in three dimensions, we can obtain volume, speed, and other factors, by assessing the visual two dimensional area beneath the curve of a function.

This information is abstract, but incredible, and you don’t need to know the numbers of how to do it, just that it can be done and should be and why.

I mean really… You know a wheel rolls on an axis, but no one expects you to be able to change a tire every day. You have to be able to figure out how to do it once in a great while.

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