The Riemann sum is a method of approximating the area under a specific curve in a graph. This is done by filling the space under the curve with multiple shapes.
To make these shapes, you need a width and height. The width is determined by the distance between two x-values. The smaller the width, the more accurate the approximation is. The height is determined by the distance between the y-value of the curve and y=0 (the x-axis). The height is measured based on the type of Riemann sum being used, some types of Riemann sums being more accurate than others.
There are 4 types of Riemann sums, the right riemann sum (RRS), the left riemann sum (LRS), the midpoint riemann sum (MPRS), and the trapezoidal riemann sum (TRS). All of them have different levels of accuracy depending on the curve of the graph, some being over-estimates, some being under-estimates.
In a Right Riemann Sum, the height of each shape is equal to the value of the function at the right endpoint of its base.
In a Left Riemann Sum the height of each shape is equal to the value of the function at the left endpoint of its base.
In a Midpoint Riemann sum, the height of each shape is equal to the value of the function at the midpoint of its base.
As stated before, when doing Riemann sums, the number of rectangles used to approximate the area determines the accuracy of the approximation. The number of rectangles is called a subdivision and is generally represented by the variable n.
When the width of the rectangles is smaller, the approximation becomes more accurate. The value of this width is generally represented by the symbol delta x (Δx).