The integral of F (positive function) from A to B is the area below the graph and above the x axis. It is calculated by approximating it with rectangles in increasing numbers. On the right, the approximation of integral of x → √(1−x²) from −1 to 1 (1.570796… = π/2).
I F is not always positive, then the areas of rectangles are signed: positive/negative if they are above/below the x axis: in the case shown below on the left, the integral between -1 and 1 of f1 is > 0, that of f2 is < 0.
In the case shown below on the right, the integral between -2 and 2 is 0.
  
   

This script allows to calculate the integral of polynomial functions.

•  The oriented area above left, where f1(x) = −x²+1:

I take 1.333… = 1+1/3 = 4/3.

• [−3, 5] 5 + 2·x − 3·x² dx

I take  −96.

We can draw the graphs with JavaScript; for example, see x21  (see the code;  se also here).

The graph, in PHP, here.

The graph, in R:

source("http://macosa.dima.unige.it/r.R")
g = function(x) -3*x^2 + 2*x + 5
BF=4.5; HF=3   # base, height of window
graphF(g, -3,5, "brown")
integral(g, -3,5)
# -96

The graph with Desmos: