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Consider the function $y=2x$. The graph of this function is here.

Next, Consider $\int 2x dx=x^2 + c$. Here is the graph.

My question is : What exactly did I do on the straight line $y=2x$ that I landed up with a parabola $y=x^2+c.$

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    $\begingroup$ I don't know why but WA computed the arc length between the two limits you gave. $\endgroup$ – Claude Leibovici Sep 1 '14 at 10:06
  • $\begingroup$ I think the question is too vague to be answered here. If what you're looking for is the definition, you'd better see Wikipedia. $\endgroup$ – tomasz Sep 1 '14 at 10:12
  • $\begingroup$ @tomasz, I do think it is very relevant here !! $\endgroup$ – creative Sep 2 '14 at 5:04
  • $\begingroup$ @Abstraction: As I have said, a question about outright definition (and a well-known one at that) can easily be answered by a glance at Wikipedia (where a detailed, illustrated and in-depth explanations can be found!), and in the current form your question is just that. If the definition is somehow unclear to you, that could be a sensible question, if you could explain how it is not clear. $\endgroup$ – tomasz Sep 2 '14 at 7:12
  • $\begingroup$ @tomasz, I am aware of the definition, but I am looking for something more in-depth visualization from the geometric point of view. $\endgroup$ – creative Sep 2 '14 at 7:36
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Finding the indefinite integral of a function results in another function whose rate of change is described by the original function you just derived. Why would this be useful? Take, for instance, you knew an equation which describes acceleration (rate of change of velocity), but instead you wanted an equation to describe the velocity of a body. If you integrated the equation again after that you would obtain an equation which describes the displacement of a body.

So if $y=2x$ was an equation to find the velocity of an object, and you knew that the object's displacement always starts at 0, you can integrate that equation to find the displacement at, say, x = 3.

Where, x is time in seconds.

Displacement:

$y = x^2 + c$

$y = x^2$ because $c = 0$ (Displacement always starts at 0)

$y = 3^2$

$y = 9$ units

Velocity:

$y = 2x$

$y = 2(3)$

$y = 6$ units$^2$

Later with definite integrals you will be able to find the area under a graph. Useful for probability distributions.

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