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In this article, we will look closer at the Riemann Sums. We will see how it can be used to estimate the area under a graph using rectangles. We will also look at both the left and right Riemann sum, and what happens when we take the average of the two.
The Idea Behind the Riemann Sums
Imagine that we have the following function:
Let us then say that we want to approximate the area under the graph in the interval: [1,3]. Let us first see the function visually:
The question is then — how can we use rectangles to estimate the area under the function on the interval [1,3]? Imagine we divide it into rectangles of equal width, where the width is equal to Δx. Before we draw the rectangles completely, let us first decide how wide they should be. We can try to decide that we want to divide it into four parts:
Let us then calculate the width of Δx. We know that our interval goes from 1 to 3, hence it has the total width of:
We also know that we have 4 sub-intervals, hence they will have a width of:
