Three Different Quantities

As the name implies, the Fundamental Theorem of Calculus (FTC) is among the biggest ideas of Calculus, tying together derivatives and integrals. One half of the theorem gives the easiest way to compute definite integrals. The other half relates the rate at which an integral is growing to the function being integrated. It is the 5th of the Six Pillars of Calculus:

The Fundamental Theorem of Calculus relates three very different concepts:

1. The definite integral $\int_a^b f(x)\, dx$ is the limit of a sum. $$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x,$$ where $\Delta x = (b-a)/n$ and $x_i^*$ is an arbitrary point somewhere between $x_{i-1}=a + (i-1)\Delta x$ and $x_i = a + i \Delta x$. The name we give to the variable of integration doesn't matter: $$\int_a^b f(x) \,dx = \int_a^b f(s)\, ds = \int_a^b f(t)\, dt$$


2. The indefinite integral is the function $$I(x) = \int_a^x f(s)\, ds.$$ That is, it is a running total of the amount of stuff that $f$ represents, between $a$ and $x$. If $f$ is the height of a curve, then $I(x)$ is the area under the curve between $a$ and $x$. If $f$ is velocity, then $I(x)$ is the distance traveled between time $a$ and time $x$.
   
   


3. An antiderivative $F(x)$ of $f(x)$ is a function with $F'(x)=f(x)$. There are actually many different anti-derivatives of $f(x)$, but they differ by constants. For instance, $x^3$ and $x^3+7$ are both anti-derivatives of $3x^2$.