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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.

  • 2
    Maybe one of the fundamental theorems of calculus could help you out: $$ \frac{d}{dx} \int f(x) dx = f(x).$$ This means that the "instantaneous slope" on the graph of $y=\text{antiderivative}$ is just $f(x)$.2012-02-05
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    @anon I think you mean $\displaystyle \frac{d}{dx}\int_a^x f(t) dt = f(x)$2012-02-05

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