Definite Integration explanation

Question

We know that definite integration provides us the area under the curve but I still don't understand one simple thing.

If we take $\int f(x)dx = F(x)+ c$

Then we say $\int_a^b f(x)dx= F(b)-F(a)$

$i.e$ how do we know that the area under the curve is difference of values of the integrated function

That is the Fundamental Theorem of Calculus. There are many routes to proving it and understanding it - answering your "how do you know" question. See https://math.stackexchange.com/questions/1991575/why-cant-the-second-fundamental-theorem-of-calculus-be-proved-in-just-two-lines/1991585#1991585 — Ethan Bolker, Feb 07 '21 at 12:37
Related thread: Why does the fundamental theorem of calculus work? — littleO, Feb 07 '21 at 12:43

score 1 · Answer 1 · answered Feb 07 '21 at 14:06

I think that instead of viewing it as "the area under the curve", you should view it as "an infinite sum of infinitesimal values". Using it to find the area under the curve is just one usage. Thinking of it as an infinite sum of infinitesimal values makes the process make more sense.

Let's start with a function:

$$ y = F(x)$$

Now, let's differentiate it without dividing by $dx$:

$$ dy = f(x)\,dx $$

$dy$ is the infinitesimal change in $y$. Now, if our graph is smooth and continuous, then the total change in $y$ between two $x$ values is just going to be the sum of all of the infinitesimal changes between those two places.

So, we can write:

$$ \text{total change in }y = \int_{x = a}^{x = b} dy = \int_{x = a}^{x = b} f(x)\,dx $$

Now, interestingly, the "total change in $y$" is actually the same as $y$, but possibly offset by a constant. That is, if we graphed the "change" in $y$ from a given starting point, it would give us the same graph as $y$ itself, just with a possible vertical offset.

Therefore:

$$ \text{total change in }y = y + C = \int dy = \int f(x)\,dx $$

So, if we just pick two pieces of this puzzle that we are interested in:

$$ y + C = \int f(x)\,dx $$

The big question is, why does the value on the right give the area under the curve of $f(x)$? The answer is that if we break the graph up into rectangles, then each one will be $f(x)$ high, and $dx$ wide. So $f(x)\cdot dx$ is the area of each rectangle, and so $ \int f(x)\,dx $ is simply the sum of all of them. The ones we are interested in are those from $x = a$ to $x = b$.

So why the difference between $F(a)$ and $F(b)$? Well, since $y + c$ is equal to the "total change in $y$", then the difference between two points in $y$ is going to give us that total change. Since we have a formula for $y$ based on points in $a$ and $b$, ($F(x)$), we can use that formula. I.e., since $y = F(x)$ (our original given),

$$ \text{total change in }y_{x = a}^{x = b} = (y + C) |_{x = a}^{x = b} = \int_{x = a}^{x = b} dy = \int_{x = a}^{x = b} f(x)\,dx \\ F(b) - F(a) = \int_{x = a}^{x = b} f(x)\,dx $$

Definite Integration explanation

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