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Riemann Sum Formula

Through Riemann sum, we find the exact total area that is under a curve on a graph, commonly known as integral. Riemann sum gives a precise definition of the integral as the limit of a series that is infinite. For approximating the area of lines or functions on a graph is a very common application of Riemann Sum formula. This formula is also used for curves and other approximations.

The idea of calculating the sum is by dividing the region into the known shapes such as rectangle, squares, parabolas, cubics, that form the region that is somewhat similar to the region needed to measure, and then adding all of the regions to find the area.

The four methods in Riemann Summation for finding the area are:
1. Right and Left methods : This method is to find the area using the endpoints of left and right  of the sub intervals, respectively.
2. Maximum and minimum methods: Through this the values of largest and smallest end point of each sub- interval.

$[a,b]$ = Closed interval divided into ‘n’ sub intervals
$f(x)$ = continuous function on interval
$x_{i}$ = Point belonging to the interval [a,b]
$f(x_{i})$ = Value of the function at at x = xi

\[\large S_{n}=\sum_{n}^{i-1}\int (x_{i})(x_{i}-x_{i-1})\]

Solved Examples

Question 1: Find the area under the curve f(x) = x2 + 1, -3 $\leq$ x $\leq$ 1 by using riemann sum with 4 intervals ?

Solution:

Given that:
a = -3,
b = 1
f(x) = x2 + 1

$\delta$ x = $\frac{b – a}{n}$ = $\frac{1 + 3}{6}$ = $\frac{4}{6}$ = $\frac{2}{3}$

Area = $left ( \sum_{i} f(x_{i})\delta{x})$ = $\delta{x}$(f(-2) + f(-1) + f(0) + f(1))

$\frac{2}{3}$(5 + 2 + 1 + 2) = $\frac{20}{3}$ = 6.6667