What exactly is integration?

Question

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 $y=x^2+c.$

Answer 1

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.