Why is indefinite integral called so? Two questions that are greatly lingering on my mind:
1.

Integral is all about area(as written in Wolfram). 

But what about indefinite integral? What is the integral about it?? Is it measuring area?? Nope. It is the collection of functions the derivative of which give the original function and not measuring area. So, why "integral"?? And what about the indefinite??? It is not measuring an infinite area ; just telling about the original functions. So, what is the logic of this name??


*Famous statement:


Differentiation breaks apart the function infinitesimally to calculate the instantaneous rate of change, while, on the other hand, integration sums up or integrates the infinitesimal changes to measure the whole change or area .
  
  Yes, totally correct but in case of definite integrals,where small changes are summed up to give the area. But how is the statement related with indefinite integral?? Do they sum up small changes??? What is the connection between them???

I am confused. Please help me explaining these two problems.

Assuming you have a function f that is integrabile in a segment, then for any value of 'a', you can define a definite integral function by: 
$$
F(x) = \int_a^x f
$$
that is defined by the area under f in the range [a,x]
Hence, we can say that all of them together (for all possible values of 'a') form indefinite integral of f - marked by
$$
\int f(x)dx
$$
Assuming f is continues in the segment, we can show - base on the derivative definition - that each of these definite integral functions is a primitive function of f (this is part of the fundamental theorem of calculus).
Next, it is easy to show that all primitive functions of f are of the form F(x) + C, where C is a constant and F'(x) = f(x).
Hence, we define  the indefinite integral of f as all primitive functions of f, and we mark
$$
\int f(x)dx = F(x) + C
$$
