Theory/Logic behind integration Let's suppose we have a curve $y = x^2$
Now when we take two points on the curve, we aren't completely sure how to connect them(will elaborate more later), compared to two points on a straight line which we are sure how to connect(shortest distance).
My argument is that the only way we know how to connect two points (precisely) is by marking the shortest distance between them (please do tell if I'm wrong here and for the sake of the question do assume this for a second) now, to connect two points on the curve we need more points between the two original points and then connect all of those points by marking the shortest distance between them. But then the same initial argument applies to any two adjacent points - that we aren't sure how to connect them again (compared to a straight line). So now we need to consider something like infinity too - basically keep increasing the resolution of the points on the curve to a non finite degree to connect the points and form the curve precisely.
Now comes my question - in this backdrop, how is it that by integrating we can find the precise value of the area under the curve?, since we were unable to form the curve in the first place precisely unless by using the concept of infinity.
Is there any literature which explains my question simply?
 A: Yes, there is an error. If you are talking about a curve like $x^2$, the error in measurement of area is proportional to h say. where h is the gap between 2 consecutive divisions you are splitting your curve to assume them to be rectangles.
Now since you are saying I'll reduce the gap between the lines to a value as close to 0 as possible, the error (since proportional) also reduces to a value as close to 0 as possible.
Recall that 10+h as Limit $h\to 0$ is $10$.
A: In real analysis the integral is usually defined using Darboux sums.
You define partition of the interval and then you let n go to infinity.
Then the sum of the (signed) areas of the rectangles (as n goes to infinity)
is defined/said to be the integral (the value of integral).      
Check this e.g.      
Darboux Integral
A: About 50 years ago a calculus instructor told us there is no direct connection between the summation series and the integral. The summation method has been used since antiquity to determine the area of the circle. I have written compter programs to do the summations, but at the end of the day it is still a guestimate. Fundamentally the best way to see this is to approach it as a ranging problem. For simplicity consider only the rising part of the curve. Reverse the process where the curve falls. If you sum the area where the left side of the rectangles touch the rising curve the summation will be understated. If you sum the area where the right side of the rectangles touch the rising curve that summation will be overstated. The limit lives somewhere between. As the width of the rectangles is decreased the difference between the overstated and understated sumations approach the limit. If you write a program to do this you can watch the limit begin to appear. BUT most computers will exceed their ability to compute fractional numbers as the rectangular slices approach zero. If you let the slice width go to 0.00001 you can probably get to within 3 units of the area limit. BUT here is the kicker: This has nothing to do with the integral, other than someone discovered that integrating the equation of a curve gives you that limit number. There is no magic connection to the summation, other than it works well enough that we have not yet discovered a significant number of places where it does not. The integration process serves as an analog of the infinite summation, but it is not otherwise related. The integral is an analog of the maximum extent, aka the limit, of the area under the curve. Taking the integral of the equation representing the long side of a triangle results in the area which is proven by calculation((base x height)/2). The same summation method can be used to approach the limit of the area under the long side of the triangle, with similar results. So taking the integral of a line equation results in the area, whether the line is straight or a curve.
