The definite integral
Every Riemann sum is only an estimate. Is there one exact number they all close in on, and what does it mean when part of the graph lies below the axis?
Where you are 3 inside
What this area is about
Rectangles under a curve give an estimate of an area or an accumulated amount, and thinner rectangles give a better one. When the Riemann sums approach a single number however the sample points are chosen, that number is the definite integral \int_a^b f(x)\,dx. It is exact, and it counts area below the x-axis as negative.
How these fit together
Definition. The definite integral as a limit of Riemann sums defines the integral as the limit of the sums as the number of pieces grows without bound, and computes a few exactly that way.
Meaning. Signed area and integrals from geometry reads the integral of a function continuous on [a, b] as area above the axis minus area below, and evaluates integrals exactly from triangles, rectangles and circles.
Results. Properties of definite integrals gives the rules every Riemann sum obeys, such as \int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx, linearity, and comparison bounds, so new integrals can be built from known ones.
A path through it
Start with the limit definition, which gives the other two their footing. Then signed area, and then the properties, which use the conventions for reversed limits that signed area sets up.
Next comes the fundamental theorem of calculus, which evaluates these integrals without sums, and then net change in context and improper integrals.