What does integration do? I know that integrals are used to compute the area under a curve. Let's say I have $y = x^2$. It creates smaller rectangles and then add up the sum (assuming that rectangles are going infinitely in number and is like going to a limit).
But I recently encountered a problem in my mind. Suppose we have a function, $y = x^2$. If we integrated it, we simply get the anti derivative of it which is $x^3/3$, assuming that the area is not of concern. What is the correlation of $x^3/3$ to $x^2$? I mean, it simply likes transforms a function into another function, but I can't get a clearer picture. When we graph $x^2$ and $x^3/3$, there is no connection visually. They are simply different graphs.
Thanks and I hope your comments can clear up my mind.

It may be helpful to divorce (somewhat) the ideas of the integral and the antiderivative in your mind.  The the definite integral is simply the concept of signed area.  The integral exists, even for functions with breaks, corners, and other points generally considered 'not nice' (provided there are only countably many such points.
The antiderivative, on the other hand, is a function $F(x)$ such that $F'(x) = f(x)$.  
The connection between them is the First Fundamental Theorem of Calculus, namely that: $$F(x) = \int_a^x f(t)dt$$ Without the fundamental theorem, there is no connection between antiderivative and integral.  While this way of thinking is perhaps a bit extreme, it provides a good way of thinking integral and antiderivative in a more rigorous way.
