Editing Riemann integral (section)

== Overview ==

Imagine you have a curve on a graph, and the curve stays above the x-axis between two points, a and b. The area under that curve, from a to b, is what we want to figure out. This area can be described as the set of all points (x, y) on the graph that follow these rules: a  ≤  x  ≤  b (the x-coordinate is between a and b) and 0 < y < f(x) (the y-coordinate is between 0 and the height of the curve f(x)). Mathematically, this region can be expressed in [[set-builder notation]] as
<math display="block">S = \left \{ (x, y)\,: \,a \leq x \leq b\,,\, 0 < y < f(x) \right \}.</math>

To measure this area, we use a '''Riemann integral''', which is written as:
<math display="block">\int_a^b f(x)\,dx.</math>

This notation means “the integral of f(x) from a to b,” and it represents the exact area under the curve f(x) and above the x-axis, between x = a and x = b.

The idea behind the Riemann integral is to break the area into small, simple shapes (like rectangles), add up their areas, and then make the rectangles smaller and smaller to get a better estimate. In the end, when the rectangles are infinitely small, the sum gives the exact area, which is what the integral represents.

If the curve dips below the x-axis, the integral gives a '''signed area'''. This means the integral adds the part above the x-axis as positive and subtracts the part below the x-axis as negative. So, the result of <math>\int_a^b f(x)\,dx</math> can be positive, negative, or zero, depending on how much of the curve is above or below the x-axis.