Question

Hi, my question is more theoritical. I know how to integrate ( mostly) but I don't understand the integration process itself. I asked teachers, looked on the internet, etc... The concept that FTC 1 and FTC 2 explain. I do not understand how integrating a function gives its anti derrivative ( therefore, I'm guessing integrating means finding the area, then how does the anti derrivative define the area).

By the logic above, a change in x(or delta x ) is the sum of the infinitesimal changes dx . It is also equal to the sum of the infinitesimal products of the derivative and time. This infinit

Answer 1

It is also equal to the sum of the infinitesimal products of the derivative and time. This infinite summation is integration; hence, the integration operation allows the recovery of the original function from its derivative. It can be concluded that this operation works in reverse; the result of the integral can be differentiated to recover the original function

Answer 2

Wikipedia explanation of the FTC and integral

Answer 3

Or you could watch session 52 where FTC1 & 2 are proved.

Answer 4

It is not infinity. Not all sums of infinite terms gives you an infinite numer. In fact, there are a lot of infinite sums that gives you a finite number. For example if you take \[\lim_{x \rightarrow \infty}(1+1/n)^n=e\] lets check: This expression is the sum of infinite terms but it does give us an finite number. This is one of the wonders on math. I hope that my answer proves usefully.

Answer 5

And Integrals is like sum all height times base. And derivatives is like height divided by base. So you can think in terms of the same reason that make multiplication and division inverse process.