Intuitive explanation for integration

Hello high school student here that has never taken a "formal" calculus course, however I know some of the basics like differentiation and limits. I'm currently reading a book on electromagnetic theory that is mathematically intensive. I seem to come across integration notation quite often but lack an intuitive understanding to be able to apply it to the physical world or in this case to understand what is going on. I have read online explanations and it seems that they discuss finding the area under the curve, so on and so forth but lack a fundamental explanation. I would appreciate if someone could explain integration to be able to gain an intuitive understanding of it which would enable me to apply it to physical concepts.

• Computing mass from density is a good example to keep in mind. It would be easy to compute the mass of an object if its density were constant. So chop the object up into pieces so tiny that the density is approximately constant for each piece. Add up the masses of all the little pieces. – littleO Sep 2 '14 at 1:53
• You might start by reading a bit about Reimann (Sp?) sums. Essentially, you create rectangles between your two bounding curves (usually between the x-axis and some function of x), find the areas of the rectangles and then sum those areas. This produces an approximation of the area below the curve. The width of each rectangle you can consider delta x. Take the limit of these sums as delta x approaches zero (as your rectangles get very skinny). That is essentially - roughly speaking - what you are doing with integration. – 123 Sep 2 '14 at 1:55
• An Intuitive Introdution to Calculus that I wrote once for my students might interest you. – davidlowryduda Sep 2 '14 at 3:06
• I upvoted this question because I like it. The reason is because now as a smart person, I have the natural desire not just to make computations but also slowly and deeply think what's going on and ask myself the question of whether I can solve a problem by a specific method. I actually like the idea of insisting on checking what the integral of a function actually is using the method of making computations directly from the definition of an integral and not just from the theorems that an integral is an antiderivative and an antiderivative is unique up to addition of a constant function. Take – Timothy Nov 5 at 0:12
• an elementry function such that if you blindly apply the rules that its integral is its antiderivative, you find that its integral is not elementry. It's up to you to decide whether to consider the result you got meaningful or not. However, one thing we can do for the original function is prove how to express it as a power series although it might have a finite radius of convergence. You don't even have to use differentiation to determine the power series. You can instead find it using the rule for finding the power series of a composition, an exponential, a product, a quotient, and a – Timothy Nov 5 at 0:16

"Area under the curve" only works if you have been trained (or trained yourself) to associate the area under some curve with the phenomenon you're trying to understand. This works fine as a visualization of the mathematics of an electromagnetic effect if you already understand that effect mathematically. That does seem a bit backwards.

In other words, I think you have an excellent question.

You have gotten at least one good answer. The following is (or should be, if I explain it correctly) completely consistent with the other answer(s), just from a slightly different perspective.

Consider a physical property such as the charge in a capacitor. We start with a capacitor with a certain amount of charge on each side and discharge it slowly through some device. While it's discharging, we keep track of the remaining charge on one side; let's say we choose the side with a negative charge. As you may imagine, we can consider the charge to be a function of time, which we might name $Q(t).$ The derivative of that function with respect to time, $\frac d{dt} Q(t),$ is the rate at which the charge changes (positive in this case, since charge is getting less negative), that is, the current, $I(t),$ flowing into that side of the capacitor at each instant in time.

Now, the integral of a function is often called an antiderivative. If you can understand how a physical quantity (for example, $Q(t)$) relates to its derivative (for example, $I(t)$), then you can work that relationship in reverse. Instead of measuring $Q(t)$ as a function of time and computing $I(t)$ from it, measure $I(t)$ as a function of time and compute $Q(t)$ from it. If you know the charge at all times, you can reconstruct the measurement of current; if you know the measurement of the current at all times, you can (in a sense) reconstruct the charge. More precisely speaking, you can reconstruct how much charge was added or removed since the instant when you started measuring the current.

Sometimes you are measuring a quantity as a function of something other than time, but the principle is the same; all that changes is the variable or dimension over which you take your derivative.

So the name "antiderivative" is very apt: given a function, $f,$ that is a derivative of some physical quantity, we take the antiderivative (integrate $f$) to reconstruct (at least partly) the original physical quantity, which is itself a function, $F.$

Sometimes integration is done over an area or volume. Again, this is just reversing a differential/derivative, but now the derivative is defined over multiple dimensions instead of just one, and it is often thought of as some kind of density.

And we should expect that the tools of calculus will mostly apply relatively directly to physical processes, since that's a very large part of the reason they were first developed.

Suppose you have some process that, over time, adds to (or subtracts from) a physical quantity. For example, a moving object has velocity, and over time that velocity results in a change in position. Or, if you apply a [variable] force to an object through a distance, over time you exert more and more work to move the object.

In simple cases, one can compute the total amount of [quantity] by multiplying: if the velocity is constant, then $d = rt$. If the force is constant, then $W = Fd$. But what if the process does not proceed at a constant rate? That is one of the problems that integration addresses. So if your force changes over time, you can imagine approximating the force over some small interval of time by a constant (since presumably over a small interval, the force doesn't change much). Do that over the entire range of times you're concerned with; over each such interval it's easy to approximate the work, since the force is roughly constant. So the total work is approximately the sum of each of those easy-to-calculate pieces. Now let the small intervals get smaller and smaller. At least if the force function is reasonable, the sum of those pieces of work gets closer and closer to the total work performed. That total work is the integral of the force over time, and the areas that successively approximate it are various Riemann sums on that time interval.

Just meant as info***

I wanted to post this link here for you without it getting lost among comments:

Understanding Riemann sums, and what happens with Riemann sums when you evaluate the limit as delta x tends toward 0, is a great way to build a more intuitive understanding of integration.