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When taught in school, we are simply told to think of $\int_{0}^{x} dx$ as open and closing brackets, for example $\int$ is equivalent to ( and $dx$ is equivalent to ). However, I know there is more to this as I have seen people add $\int$ into equations such as $dx = dt$ and get $\int_{0}^{x} dx = \int_{0}^{t} dt$ when deriving physics equations.

Now, my question is basically this. I know $dx$ stands for delta, as in $x_1 - x_0$.

But is $\int_{0}^{x} f$ the same as saying $\sum\limits_{i=0}^{x} f$

That is, the sum of $f(i)$?

Thus, when multiplied with $dx$ it becomes the definition of a definite integral?

Is this right? Can we think of the definite integral symbol as nothing more than an summation or for loop?

Aka,

I know that this $\sum\limits_{i=0}^{x} f$ is the same as this for loop here:

for( int i = 0; i < x + 1; ++i ) {
    sum += f(i);
}

But are these also equal to $\int_{0}^{x} f$

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  • $\begingroup$ Example: In the sum $\sum_{t=0}^{x}t$, the values of $t$ are discretized. In the integral $\int_{0}^{x}t\mathrm{d}t$, $t$ takes all the values in the interval $[0, x]$. Take some examples with your $f$ to see the difference. Also see this post $\endgroup$ – Jika Jun 2 '14 at 23:27
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The symbol $\int$ is from Leibniz (I think he was the first to use it), and he called it a "summa", or "sum". The term "integral" came from the idea of "the whole", or "putting all together" - in other words, a "sum" (Leibniz wanted to discard the term 'integral' in favor of his 'summa', but it never happened).

So you are correct that the symbol $$\int_a^b f(x)$$ represents the sum of all values $f(x)$, where $x$ ranges across the interval $[a,b]$.

Note the stress on all. Adding "infinitely many numbers", for Leibniz, resulted in an infinite number. The integral above is therefore infinite in value.

The reason this type of notation is hardly ever seen, is that mathematicians often considered differentials by themselves, such as $$dx$$ which is infinitely small, but functions which were integrated ("summed") almost always contained a differential, as in $$\int_a^bf(x)\,dx$$ So the infinitely small occurred frequently, but the infinitely large hardly at all. As an exception though, consider this statement of Johann Bernoulli: $$\int z=\frac{zz}{1\cdot 2\cdot dz}$$ which in modern notation is just $$\int z\,dz=\frac{z^2}{2}$$


A few notes are in order.

First, your expression $\sum\limits_{i=0}^{x} f$ is not meaningful - however, your code seems to suggest that you have in mind some kind of discrete summation, and this is not what the integral represents. It is a continuous summation, one that includes all values of the function over the domain, not just over integers in the domain.

Second, the notation $\int_a^b f(x)\,dx$ means something different today that what it used to mean. It is essentially a limit which has a certain value, and writing the expression without a "$dx$" would not represent anything at all, by today's standards.

Further to that, your teacher's explanation is not technically incorrect, but it does not seem very insightful - for your intuition, it is (in my opinion) better to think about what the symbols actually represent than that sort of "symbolic-mechanical" understanding.

Source: H. J. M. Bos. "Differentials, higher-order differentials and the derivative in the Leibnizian calculus." Archive for History of Exact Sciences 14.1 (1974). Link.

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