Integrals, Integrands, and Understanding the Notation

Even though mathematics is one of the subjects I enjoy studying, it’s not always easy. Nor does everything always make sense to me. One of these things was the notation for an integral.

If you’re reading this, you probably have an idea of what an integral looks like. The simplest one looks like this:

You recognize the $f(x)$ part, but the other two parts aren’t components you’ve seen before. When I first learned about integrals, I don’t think I really internalized the notions of the two other symbols, and it was to my detriment. In mathematics, if things look funny, it’s probably because you don’t fully understand them. The reason is that ideas are made to be as compact and helpful as possible in most mathematical concepts that you will see in your education. Said differently: complexity isn’t introduced for complexity’s sake.

The symbol on the left ($\int$) is just the sign of integration. Much like we write $\frac{d}{dx}$ or $f’(x)$ to denote derivatives, the sign of integration is there to tell you that an integration is being performed. However, the shape of the symbol isn’t arbitrary. The symbol is a sort of horizontally narrowed and vertically stretched out “S”, and it stands for the word “sum”. After all, integrals come from the idea of taking the sum of a bunch of tiny pieces of a function, and so the sign of integration is supposed to be a nod to that.

However, it’s the other symbol which has much more meaning. The $dx$ part of the integral is what signals to the reader what variable will be integrated. In my example, $x$ is the variable being integrated.

When I first learned about integrals, this was where I stopped digging deeper. I more or less just went ahead and calculated, only using the $dx$ as a small reminder that I was supposed to be interacting with respect to $x$. This approached worked for me, but it definitely wasn’t the best I could have done. Instead, here’s the much better way of thinking about the $dx$ part.

First of all, $dx$ is called the infinitesimal or differential of $x$. These are just long words to say that the portion of $x$ we are considering is really tiny.

You also probably know that a way to interpret a definite integral (one that has bounds of integration) is to think of it as the net area “under” a curve between two points. That’s the “physical” way to look at it, and it has a lot of meaning for our notation of an integral.

What kind of units does an area have? It has units of a length squared, such as $m^2$. Therefore, if our integral is just a bunch of addition of tiny “chunks” of area, each chunk needs to have units of area.

If we look back at our integral:

We can imagine the term $f(x)$ being considered as one of the lengths in question. So what’s the other one?

It’s the small portion $dx$.

The reason this was hard for me to grasp at first was that we were discussing $dx$ as though it was so small that it was basically not there. As such, I didn’t think of it as a “real” quantity. It was just a symbol that I put inside my integral, and it had no actual meaning. However, I couldn’t have been further from the truth. By taking $dx$ to have a geometrical meaning, the idea of an integral suddenly felt so much more grounded to me. The $dx$ wasn’t some ornament. It was an important part of the integral.

And that’s what I want you to get from this. The differential $dx$ is very important in concepts later on, so it’s a good idea to get a firm understanding of what that differential actually means.

Understanding Algebra and Balancing Equations

If there’s one thing I’ve learned about mathematics during my many years at school, it’s that having a solid foundational understanding of the main components can go a long way towards learning new subjects within mathematics. Unfortunately, this is what is often lacking for students, and it can have the knock-on effect of making later concepts more difficult to grasp.

As a tutor, the concept I see students struggling the most in secondary is a basic understanding of algebra and how equations work. This becomes evident to me because the concepts I’m supposed to help students with aren’t necessarily knowing how to manipulate equations, yet this is exactly the aspect they struggle with. I feel like most of the students I help don’t have a solid grasp of what it means to have an equation (or solve a system of equations), which is why I wanted to go over the common mistakes I see.

Mistake One: Flipping Signs

Here’s an equation:

If you want to solve this, the step-by-step method involves bringing the 7 over to the other side, followed by bring the 5 over as well, giving us an answer for $x$. However, the mistake I often see here is one of the following:

  • Bringing the 7 to the other side like this:

  • Or bring the 5 over like this:

In both of these cases, the mistake is one of signs. I remember when I was first learning this, I would try to simply remember that multiplication and division doesn’t include flipping signs when “crossing” over to the other side of an equation, and that I would have to flip signs if it was addition or subtraction. This is how I learned it at the beginning, but it definitely wasn’t a good method.

The reason is that mathematics is logical. If you can’t figure out why something ends up being the way it is in an answer, it’s because you don’t fully understand it. In this case, I was content to just remember my rule for flipping the signs, but that made me disregard real reason why my rule worked. Now, I have a better understanding of how these equations work, and so I don’t have to remember my rule.

I think the trouble stems from the way we talk about these equations. When we’re solving for $x$ as in the above equation, we talk about “bringing” numbers to the right-hand side in order to isolate our variable. However, the word “bringing” is pretty vague, and ends up having a different meaning depending on if we are talking about the 7 or the 5 in our example. What we really mean when we say, “bring 7 to the other side” is to subtract 7 from both sides of the equation, like so:

From here, it’s much more clear what’s going to happen. The 7s on the left-hand side will cancel, and the one on the right will be subtracted from 2. There’s no need to think about flipping signs. Instead, it’s all about what it will take to get the seven away from one side of the equation.

It’s also important to note that this works because you’re doing something to both sides of the equation. I always find it’s useful to think of an equation as a balance. Therefore, if you add something to one side, you have to add the same amount to the other if you want the balance to remain in equilibrium.

Likewise, once we have isolated $5x$ and want to find out what $x$ is, we simply need to divide both sides of the equation by 5. When thinking about equations this way, it makes perfect sense that you won’t be flipping the sign, since you want to isolate $x$ through dividing both sides by 5.

Mistake Two: Not Following The Order of Operations

This one’s more common when students are first learning, but it’s still something I see quite often. Let’s return to our example:

Sometimes, I see a student do something along these lines:

To the student, this makes sense, since I they want to “bring” the 5 to the other side of the equation. Unfortunately, this disregards the 7, which is a problem. To help them see this, I ask, “What did you do to bring the 5 to the other side?” Usually, the response will be something along the lines of, “I brought the 5 to the other side, so it’s now a division.”

Once again, this shows the danger of using language like “bringing over”, because it masks what the student is actually doing. Once I tell them that in order to “bring” something to the opposite side of an equation, they have to do the same operation on both sides of the equation, I show them how that includes dividing the 7 as well. It’s not necessarily intuitive to them at first, but once I get them on board with the idea of operating on both sides of the equation, they begin to see how they cannot just transport a number to the other side and do the inverse operation. They have to do it on both sides. This also shows them how isolating for a variable and then dividing or multiplying each side usually avoids this common mistake.

These are just a few mistakes I’ve seen through tutoring students and getting them used to manipulating equations. It’s not always easy for me to see their difficulties, since years of practice makes manipulating equations for me an easy process, but I try to find concrete arguments and reasons to show them where their mistakes are and why they are mistakes. It can be horribly confusing to be told that your answer is wrong, while not being given any explanation. As such, I try to be mindful that the best way to make students understand their mistakes is to show them situations in which their mistake obviously makes no sense. From there, I can lead them to becoming more comfortable with algebra and equations.

Can You Explain It?

Despite needing to compute many quantities in my various physics and mathematics classes, I like to take a slightly different approach to studying for a test. First, I will do a bunch of practice problems in order to strengthen the “muscle memory” of how to do certain types of questions. This usually means going over assignments and doing examples. This particularly helps with refreshing my mind for questions I haven’t done in a few weeks.

However, this isn’t the complete extent to which I prepare for my tests. When it is the night before my test, I make sure to review concepts instead of specific problems. What this means is that I won’t be doing actual calculations. Instead, I’ll make sure I can explain the steps that I need to do in order to solve certain questions. What this does is refresh my mind to the types of questions that may be asked during the test, and I find this gives me the confidence that I know what I am talking about. While some may simply do the exact homework problems again, I want to make sure that I can adapt to something different on the test. My hope is that doing this will get me thinking about the concepts in general.

As such, I’d suggest trying the same during one of your tests. Don’t just do the practice problems and call it a day. Make an effort to be able to explain what you’re doing and the things you should be looking for in order to solve certain questions.

Full Explanation

What is the one thing in mathematics or physics you feel completely comfortable with? In other words, which concept do you find you have grasped to such a degree that you’re able to get a good grasp on any problem concerning it? For myself, the mathematic concept that I feel pretty comfortable with is the idea of tangent lines and planes to certain functions (multivariable or not). I’m not saying that I understand a problem within three seconds, but I’m generally able to figure out the solution without too much difficulty. In physics, the concept I’m comfortable with is waves. I’m able to write equations for the wave-like motion of various objects and phenomena, and I’m good at extracting information out of them. Therefore, that would be my most comfortable area at the moment.

(Of course, it may come as no surprise that I’m comfortable with these two areas because I’ve recently taken a class on them. As such, it’s not exactly surprising that I’m used to them.)

However, there are plenty of concepts which I am not familiar with. In mathematics, I always get caught up when I have to deal with chords in a circle or if I have to use hyperbolas and other not-frequently used functions. In physics, the concepts of energy, torque, moments of inertia, and other physics related to rotating bodies can often throw me for a loop.

The reason, I think, is that I never had a full, thorough explanation of most of these concepts. As such, I’m always operating on a level that includes a bit of knowledge, but not necessarily one that makes me confident that I know what to do. In other words, I have a shaky foundation. And, as we all know, a shaky foundation has consequences for later on. When we aren’t comfortable with a subject, the problems will only resurface in other concepts later on.

To combat this, I’m setting myself a goal of filling the holes of my knowledge and finding those full explanations that I might have missed at different points in my education. That way, I’ll be improving my foundation so that I can actually move forward and not be so confused with concepts.

If I can offer one piece of advice, it is this: don’t settle for half-explanations. When learning, make sure that you understand the concepts to a point that you are completely comfortable with the ideas. If you have questions, make sure you ask them. Don’t worry about seeming “dumb” or not knowing enough. What’s more important is that you understand, and nothing else.