Calculus&Mathematica is pleased to acknowledge
the heavy influence of Emil Artin's book  
Calculus with Analytic Geometry (notes by G. B. Seligman),
Committee on the Undergraduate Program,
Mathematical Asociation of America, 1957.
Although dated in spots, this short book remains a jewel.

Here is the idea of the integral:
Take a function and plot it on an interval .
It might look something like this:

If any picture is missing, then execute all the blank cells below.

Look at the area between the graph of and the -axis:

If a part of this area lies above the -axis, then call it positive.
If a part lies below the -axis, then call it negative.
This gives the signs as shown below:

Now define a number that all folks call the integral of from to .
You signify this number by writing
       
and you calculate its value by summing up the measurements of these areas taken with the corresponding signs.
Accordingly, if are the points between and at which the curve crosses the -axis as shown below:

To get a quick review of this, grab all four plots and everything between them, animate, and run forward at speed 2 or 3.

Then you calculate
       
by taking the sum of four measurements:
       
where
        and are positive
and
        and are negative.

B.1.a)

B.1.b)

Make the indicated area measurements:

B.1.b.i)

B.1.b.ii)