Question regarding dx being infinitely small In the book "Calculus Made Easy",there is a chapter on smallness of different degrees.There it was said that it was okay to treat high powers of a small fraction negligible and can be ignored.They took $dx$ to mean numeracally $\frac{1}{60}$ or $\frac{1}{1000}$ for example and discarded their high powers as relatively small to the other terms and ignored them.Now why do we need to take $dx$ infinitely small when we can take $dx=0.01$?In the end if we have something like $1000000060$ we discard the $60$.Then what is the need to take $dx$ infinitely small?Same goes for integration.If we can ignore the relatively small area,then we get the perfect area.What is the need to take $dx$ infinitely small?We can for example take $dx=1/64$ and can go with this forever.
I have given a picture for examples.
 A: It may be worthwhile to repeat here what @fleablood wrote in a comment (I hope I'm not mis-representing anything). Calculus is not about neglecting small quantities (relative small, or absolutely small, or any kind of small). Calculus is about computing precisely. Very precisely. It's about computing things that often involve some sort of infinite approximating process. This is often codified in the form of a limit. The language often employed is dynamical (e.g., $x$ vanishes, or tends to infinity). That is so for historical reasons and in order to appeal to a certain intuition that the formalism is trying to capture.
So, calculus is not the art of neglecting small quantities. It is the art very precisely quantifying how a quantity changes in relation to another quantity, with an emphasis on quantifying the effect of 'small' changes. The difference between small and large here is a question of perspective. It guides the questions we ask. For instance, asking if a given function has a local maximum is a question about the behaviour of the function in the very small scale. Asking if it has a global maximum is a large scale question. The tools of calculus are geared toward answering the former, not the latter (of course, we often answer the latter in a calculus setting, but the process of doing so is distinctly different than the classification of local extrema).
Now, this is from the perspective of pure mathematics. One may argue that from an engineering perspective one can (?) set a threshold of smallness and just discard things if they are small enough. Famously, for an engineer, $\sin(x)=x$ for all $x$ small enough. But it takes little effort to improve this and write the correct statement $\lim _{x\to 0}\frac{\sin x}{x}=1$ and then proceed to express $\sin(x)=x+r(x)$ where the remainder term satisfies $\lim _{x\to 0}r(x)=0$. This is the first order approximation of $\sin (x)$ expressed precisely and the remainder can further be analysed to produce more control over the approximation. In other words, we do not neglect $r(x)$. Rather, we replace it under control with something else, easier to compute, but we make sure not to lose track of our approximations. After all in the absence of further specification, $17$ is just a good an approximation of $\pi$ as any other value.
