Three Different Concepts

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 integral function 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 $3x^2$.