Intuition Behind the Concept of the Integral

It's something we've all seen. Throughout high school and perhaps even university, a strange S-like sign terrorises many pupils. Integrals, like derivatives, appear often in physics and mathematics. Many individuals learn how to apply it in a statistical framework, but they ignore the intuition and underlying logic that underpins it.

What exactly is an integral? Let us just pause a second to review addition when we get started on this topic. Whenever we introduce a finite number of integers, things get very simple. 3+3+3 Equals 9 and 1+1 = 2. If we have to add a lot of integers, it can get a little boring since we have to type the operation sign "+" over and over. As a result, a shorthand notation for addition is required. Fortunately, such a symbol actually exists, and it employs the Greek uppercase Sigma, or?. Take a glance at the illustration above. A succession is denoted by the notation a(k). A series is an ordered collection of related data, for those who don't recall. It may be thought of as a function with the positive integers as its domain. For instance, in this scenario, a(k) = 5, -1, 10, 305, 2, -1, with a (1) = 5, a (4) = 305, and etc. Aside from the sequence, a summation requires a beginning and ending index to determine where the summing should begin and conclude. Glancing at the diagram again, we can see that we need to append the terms of the series a(k) between index k = 1 to index k = 5. When we employ the above-mentioned sequence a(k), we get the following result: 5 + (-1) + 10+ 305 + 2 = 321. While we have to combine an endless amount of numbers, the problem becomes more difficult. This type of sum is referred to as a Series. In mathematics, a Series is the sum of an infinite number of discrete values, indicated by the Greek uppercase, or whose end index is plus infinity.

Concept of Integral

There isn't much of a difference in notation between this and the finite sum we studied before. You possess the sequence, the first index (n=1), and the last index, infinity. You could see how the above sum might begin by replacing different values of n: 8+ (— 4) + 2.666 + (— 2) +... While working with endless sums, the most challenging component is calculating the final result of the summation. Many mathematicians have devised innovative methods to solve some distinctive infinite sums. When we discuss about Series in maths, it's important to remember that we're working with finite things. We have such a series, therefore we put in all the natural numbers to add its parts. But how can we apply this concept of multiplication to the continous world? How would we sum all of a function's potential values?

At first sight, it appears like we won't be able to. For any real x, examine the function f(x) = x. The result must be infinite if we combine all of its values for all positive real x's, right? We're essentially combining all of the positive real numbers together. We can see right away that trying to apply the notion of a Series to the continous world causes issues. This is why we approach the issue in a unique way. Rather than summing all of a function's values, we look for the entire area underneath its graph! Determining the complete area may appear strange at first, but it is the best we can do. Consider that for a moment. In a broader sense, we need to have an integer to "quantify" the sum of all a function's values. Things were straightforward when we were working with sequences. We just include all of their words. In the case of real functions, however, this is not possible. As a result, the overall amount underneath the function's arc is the quantity that comes closest to this intended quantified. Let's just see how we can make this happen. Consider the curve of such a real function f. (x). To begin, we split the area of the function graph into rectangles of constant cross section x and height f(x) — the function's value at x — as shown in the diagram below.

Clearly, the total of the rectangles' surfaces is not a suitable estimate for the function's curve. Therefore, what are our options? Keeping the rectangles smaller and smaller, making x smaller over time, is a wonderful concept. If we do this, we obtain the image below.

One can easily see it as x decreases, the estimate of the area improves. To calculate the overall curve, we reduce x to zero and add together all the rectangles that result.

That is all there is to it! At least in principle, we have discovered the approximate value of the function. We now need a technique to quantitatively characterise the entire operation.