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sorry if this has been asked before, but I can’t seem to find my question in particular.

Anyway, in the Second FTM it says $$F(x)=\int_a^xf(x)dx$$

If I understand correctly is just the area under the curve. no problem there.

Then it says that$$\int_a^bf(x)dx=F(b)-F(a)$$

if I’m thinking this correctly, it would make sense because in $F(a)=\int_a^af(x)dx$

the area is $0$ so I’m just left with $F(b)$ which is the integral from a to b, this is where I think I have to be wrong, because in every example I see, they take the value of $a$ and plug it in $F(x)$.

For example in $$\int_2^5x^2dx$$ with $F(x)=x^3/3$

they take $F(5)$ and $F(2)$, in particular $F(2)=2^3/3$, but shouldn’t $F(2)$ always be $0$ because it’s basically just $\int_2^2x^2dx$?

p.s Sorry in advance for any mistake I made in formating or anything else.

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There seems to have been a mix-up in what the letter $F$ refers to in each case.

What the following equation means

$$F(x) = \int_a^x f(x)\, dx$$

is that $F$ is a *particular choice* of antiderivative of $f$; that is, $F$ is a function such that $F’=f$, and moreover $F(a)=0$ for this choice of antiderivative.

However, the following equation

$$\int_a^b f(x)\, dx = F(b)-F(a)$$

is true for *any* choice of antiderivative $F$; that is, any function $F$ for which $F’=f$ will suffice. Antiderivatives are only defined up to a constant of integration, but the constant of integration cancels out since you’re subtracting one from another.

In the first of these equations $F$ is a specific choice of antiderivative; whereas in the second equation, the constant of integration can be anything.

To resolve your issue in the case where $a=2$, $b=5$ and $f(x)=x^2$, note that

$$\int_2^x f(x)\, dx = \frac{x^3}{3} – \frac{8}{3}$$

so in this case, choosing $F(x) = \frac{x^3}{3} – \frac{8}{3}$, rather than simply $\frac{x^3}{3}$, gives you what you want.

This kind of confusion is very prevalent and the primary reason behind the confusion is the wrong definition of definite integral as area under a curve.

It is important to first understand that the definition of symbol $\int_{a}^{b}f(x)\,dx$ has nothing do with area as such. The definition has to be based on numbers $a, b$ and the function $f$ defined on interval $[a, b]$. One such definition was provided by Bernhard Riemann and it assumes $f$ to be bounded on $[a, b]$. I will leave the definition of Riemann integral to the standard textbooks of analysis and focus next on the Fundamental Theorems of Calculus.

**First Fundamental Theorem of Calculus**: *If $f$ is bounded on $[a, b]$ and the Riemann integral $\int_{a}^{b}f(x)\,dx$ exists then the function $F$ defined on $[a, b]$ by $$F(x) = \int_{a}^{x}f(t)\,dt$$ is continuous on $[a, b]$ and $$F'(c) = f(c)$$ for any point $c \in [a, b]$ where $f$ is continuous.*

**Second Fundamental Theorem of Calculus**: *If $F$ is differentiable on $[a, b]$ and the derivative $F’ = f$ (say) is Riemann integrable on $[a, b]$ then $$F(b) – F(a) = \int_{a}^{b}F'(x)\,dx = \int_{a}^{b}f(x)\,dx$$*

Note that the first FTC describes a function $F$ based on a given function $f$ which has some nice properties ($F$ is continuous on $[a, b]$ and differentiable at those points where $f$ is continuous). But this $F$ is not an anti-derivative of $f$ on $[a, b] $ because we are not guaranteed that $F'(x) = f(x)$ for all $x \in [a, b]$. We have $F'(x) = f(x)$ only at those points $x$ at which $f$ is continuous. At points where $f$ is discontinuous the function $F$ may or may not be differentiable.

The second FTC deals with anti-derivatives. It says that if $F$ is anti-derivative of $f$ on $[a, b] $ i.e. $F'(x) = f(x)$ for all $x \in [a, b]$ and further that if $f$ is Riemann integrable on $[a, b]$ then the definite integral $\int_{a}^{b}f(x)\,dx$ can be simply evaluated in terms of the difference between values of anti-derivative (i.e. as $F(b) – F(a)$).

The function $F$ used in first FTC has a different role than the function $F$ of the second FTC and it is useless to think of them as same. Things change drastically when the function $f$ (which is to be integrated) is guaranteed to be continuous on $[a, b]$. When $f$ is continuous then both the FTC get merged into one theorem which we can simply call *FTC for continuous functions*:

**Fundamental Theorem of Calculus for Continuous Functions**: *If $f$ is continuous on $[a, b]$ then $\int_{a}^{b}f(x)\,dx$ exists and the function $F$ given by $$F(x) = \int_{a}^{x}f(t)\,dt$$ is an anti-derivative of $f$ on $[a, b] $. Moreover if $G$ is any anti-derivative of $f$ on $[a, b]$ then $$\int_{a}^{b}f(x)\,dx = G(b) – G(a)$$*

And now you see the connection between the word “anti-derivative” and the integral $$\int_{a}^{x}f(t)\,dt = F(x)$$ The function $F$ is **an anti-derivative** of $f$ and not necessarily *the anti-derivative* of $f$. You also remember the fact that *a function does not have a unique anti-derivative and two anti-derivatives of the same function differ by a constant* and this is the reason for using the constant of integration while calculating *indefinite integrals*. This is also the reason that I have used a different letter $G$ for a generic anti-derivative in the theorem mentioned above and the letter $F$ denotes a very specific anti-derivative represented in the form of a definite integral.

Thus when you wish to calculate $\int_{2}^{5}x^{2}\,dx$ by the use of anti-derivative $x^{3}/3$ you are just choosing one of the infinitely many anti-derivatives available and it will work as expected. You may wish to choose the specific anti-derivative $\int_{2}^{x}f(t)\,dt = (x^{3}/3) – 8/3$ and this will also work fine.

You can also think in this manner. The anti-derivative $F(x) = \int_{a}^{x}f(t)\,dt$ is a very specific one and has the property $F(a) = 0$. Other anti-derivatives of $f$ will not have this property of vanishing at $a$. And the fundamental theorem says that the choice of anti-derivatives does not matter in evaluating the definite integral.

Here are some aspects which might be helpful

Be careful about

not mixing upin the first part of FTC the variable of integration with the variable $x$.

\begin{align*}

F(x)=\int_a^xf(t)\,dt\tag{1}

\end{align*}

The variable $t$ in (1) is abound variablesimilar as the index $k$ in a sum $\sum_{k=0}^nf(k)$, while $x$ is a free variable.The function $F$ is

notthe area under the curve. The area is anumberwhile $F$ is not a number but afunctionin $x$.But, when we

evaluatethe function $F$ at a specific value $x=b$ we obtain thenumber

\begin{align*}

F(b)=\int_{a}^bf(t)\,dt+F(a)

\end{align*}

resp.

\begin{align*}

\int_{a}^bf(t)\,dt=F(b)-F(a)\tag{2}

\end{align*}

and $F(b)-F(a)$ in (2)canbe interpreted as (signed) area under the curve in $[a,b]$.

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