The Riemann summation method is needed to calculate the approximation of area under curves. In the case of right and left methods, the approximation is performed using right or left endpoints for each subinterval respectively. There are minimum or maximum methods to give approximation with the help of smaller or larger endpoints for each of the subintervals. The values of Riemann sum could be given as the sub intervals from top to the bottom right.
In mathematics, the Riemann sum is defined as the approximation of an integral by a finite sum. It was named after the German mathematician Riemann in 19th century. The most common application of Riemann sum is considered in finding the areas of lines or graphs. It can also be used for calculating the length of curves and other approximations too.
The Riemann sum is calculated by dividing a particular region into shapes like rectangle, trapezoid, parabola, or cubes etc. Now you have to calculate the area for each of the given shapes and add them together to find the end result. This approach is used to find the numerical approximation too in the case of a definite integral even if calculus does not make it easy to find a closed-form solution.
[a,b] = Closed interval divided into ‘n’ sub intervals
f(x) = continuous function on interval
xi = Point belonging to the interval [a,b]
f(xi) = Value of the function at at x = xi
\[\large S_{n}=\sum_{n}^{i-1}\int (x_{i})(x_{i}-x_{i-1})\]
Here, errors are generated when adding up the areas of the different shapes. When shapes get smaller than usual then the sum approaches to the Riemann integrals. The basic idea behind the concept is dividing the domain into partition or pieces and multiply the size of each piece by some value of function taken on that piece or sum of all these products.