When we heard the word “integration” we usually think about the symbol with upper and sub-script numbers and we associate the term “integration” to the process of solving the whole expression.  The first thinking is incorrect since the symbol is only a representation of th
e whole operation. For the second idea, the whole
process involves a bunch of mathematical reasoning that is a combination
of antidifferentiation and integration. But thinking deeper about the concept of integration and the definition of Riemann Sums, we infer that integrate is finding the area under a certain curve using either good approximations as a Riemann sum do or a more “powerful tool”.

At this point, it is when antidifferentiation appears: it consists in looking for an expression (this means an equation) to integrate a function. Once we have the expression written as an equation, then the prior mentioned “powerful tool” is used: The Fundamental Theorem of Calculus helps us to describe the area under a given the curve. But we only explained how antidifferentiation works, not why it is important: well, the FTC requires antiderivatives to work.

In MATH 100 we have learned how to differentiate; this term can give us a hint to follow at the moment of thinking about antidifferentiation. As simply as it sounds, antidifferentiation is the opposite of differentiate. If we remember the concept of differentiation, it will come to our minds: a method to find the slope of the tangent line of a curve using mathematical equations (the power rule gives us the easiest way to obtain a derivative: reduce the exponent of a function by 1 and multiply by the exponent we had before subtracting) which can be reversed to get the first expression, but doing this means antidifferentiation which is closely related to integrals. Students usually relate these concepts to the word derivative and besides we as students did it. But the problem comes when students think that integral and differential calculus are opposite processes, they might be though. However, they conflate the concepts when they relate everything about derivatives and everything about integrals enclosed in these two words. Then everything about antidifferentiation and integration (join the both concepts in the same perception) and then they usually forget the principal idea behind integration.

Have you ever considered the volume of water that covers the Earth? Or have you seen at the sky and wonder how many stars are in certain galaxy? Many scientists have wondered things like that, in fact the purpose of science is to give an answer to phenomena in the world and simplify the concepts to make them understandable for everyone in the world.

Now, let’s consider a galaxy and some astronomers that started their observations on that galaxy or better than that, think like those astronomers; if you wanted to know how many stars would be there applying an easy method, you should be wondering where to start. Perhaps, I would calculate the area of one star and the area of the target galaxy and I would use this data to determine the number of stars. However, the area of a galaxy or even the area of a star is not calculable because there are not regular polygons. We need an approximation to describe
the area (a method that is not conventional).

Consider another situation where a group of geoscientists want to know the area of a tectonic plate but how can they measure the surface of an irregular shape? This is a challenging question because tectonic plates do not have any similarity with the common geometric figures for which we can easily calculate their area. How can you be sure about the surface of your country? It is another complicated issue that can be solved using mathematics.

Scientists and mathematicians have wondered these kinds of questions in the past and have come up with good approximations. Researchers and scientists started from previous knowledge. Since that scientists from different ages (and we will be scientists someday to join them all) began with very simple definitions, creativity was required to solve the problems they worked on.

For example, imagine yourself as Riemann (at this point is kind of weird since we already know what he did though). Let’s start by taking whatever irregular shape and divide it in small defined portions such as simple rectangles.  Now, we can make an approximation of the figure’s area by filling it with rectangles. The accuracy of the measurement depends on the number of rectangles used. The more rectangles you used, th
e more accurate your calculation will be.

Throughout the years, scientists have improved the methods to get an accurate value of areas, there is where the concept of integral appeared.