Interpretation of indefinite integral

The indefinite integral (antiderivative) $F$ of a function $f$ is defined as

$$F(x) = \int{f(x)dx}$$

without bounds. The defining properties of this function are:

1. The derivative of $F$ is the original function $f$

   $$\frac{d}{dx}F(x) = f(x)$$

2. $F(b) - F(a)$ gives the area under the curve $f$ over the interval $[a,b]$

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

Note that, unlike the derivative function (whose interpretation at a point is the slope of the tangent at that point on the original function), the antiderivative does not have an exact interpretation at a single point $F(x)$.

A resource on integrals.