Fundamentals of calculus Chapter 2: Integration

Fundamentals of calculus

Chapter 2: Integration

Integration is the converse of differentiation. To understand this better, it must be known that during differentiation an important information is lost. Let $\frac{dz(x)}{dx} = z’(x)$. Then, we cannot get back $z(x)$ from $z’(x)$ because, while taking the difference $z(x+\Delta x) – z(x)$ only the information regarding the difference between a point in Z axis and a neighbourhood of that point is retained. But the actual orientation is lost. Throughout this article, unless stated $\Delta x$ is limited to zero. To see how this could cause the loss of a piece of information, let me elaborate this with a simple algebraic example. Consider $a-b = c$ to be true. Then we may write $a – (a-c) = c$, here $b=a-c$.

Fundamentals of calculus Chapter 1: Differentiation.

Fundamentals of calculus
Chapter 1: Differentiation.

Can we really call calculus an art? Moreover, can we call mathematics an art? Most mathematicians will agree to the answer: yes. We express ourselves, our creativity through mathematics. It’s not always about calculation. Read my article ‘why do we need mathematics?’ to better understand what I believe mathematics is really. I will not agree that the explanation I had given there was enough. But anyway, I believe mathematics is an art because it is not based on mere calculations but based on creating new rules, using them to express what is around us in a more precise and defined manner. Most of the time, mathematicians would define these rules not for the sake of answering questions that follow up with our reality. But they just do it, in hope that their discovery might come in handy to some scientist in the near future. Some don’t even care about its necessity and for the sake of not expressing themselves to be someone ignorant of the reality most people scramble through physics equations or some other kind of scientific material to find the right place for their equations or mathematical rules to fit in and prove to be beneficial.

Calculus Vs algebra:-

-By K Sreram

This article was first written on 2013

Introduction:-

This article was written to show the difference between arithmetic algebra and calculus. Differentiation and integration is far different from arithmetic algebra (though arithmetic algebra is used to represent the process of differentiation, differentiated equations, integrated equations or expressions) because, arithmetic algebra does not support the results provided by differentiation or integration process. Arithmetic algebra shows that the result provided by differentiation or integration process is approximated and the error will never be zero; but calculus says exact values.

I have proved all this in this article. This article also shows us what actually calculus is and how to apply them practically. Differential and integral calculus is a modification done to the mathematics at the time when arithmetic (and some geometric) algebra was prominent. It defines a wrong result (according to the ordinary algebra) to be correct (with proper proofs).