Take a look at the image above. By breaking up the curve into chunks, and then estimating the area of each one with a rectangle, we can estimate the area under the entire curve. Such an estimation is called a Riemann Sum, and it is the foundation upon which integrals are built.
There are many ways you can calculate a Riemann sum, but each involves partitioning the curve into chunks, and finding the area of one rectangle inside one of the chunks. While it is not strictly necessary that all rectangles have the same width, we will assume for this discussion that they do; and if we partition into n rectangles, each rectangle will have the width (b – a)/n. We will discuss three different types of Riemann sums: Left, Right, and Midpoint.
Left Riemann Sum Let f(x) be a continuous function on [a,b], and let n be an integer. Define Δx as (b-a)/n. Then we can define the Left Riemann Sum by:
Right Riemann Sum Let f(x) be a continuous function on [a,b], and let n be an integer. Define Δx as . Then we can define the Left Riemann Sum by:
Midpoint Riemann Sum Let f(x) be a continuous function on [a,b], and let n be an integer. Define Δx as (b-a)/n. Then we can define the Left Riemann Sum by:
Example: The below image shows the difference between the three sums: which function you pick for each rectangle. In each sum, Δx is the width of the rectangle, and the function value is its height. Using a Left Riemann Sum with n = 4, estimate the area under the curve f(x) =x2 between x=0 and x=1.
We’ll begin by expanding the summation notation:
We then calculate each value, obtaining our final answer:
Note that the exact answer for the area is 1/3. Our answer is off by a large percentage, since we only used four rectangles. Had we used more rectangles – say, as many as 100, or even 10,000 – our answer would be much closer.
So how do we make our answer perfect? We use limits again. Imagine Δx getting smaller and smaller – even as it approaches 0. As Δx gets smaller, n (the number of rectangles) will get larger. Of course, you can’t have a rectangle with zero width. But you can take the limit as Δx approaches 0; as n approaches infinity. This allows us to present the limit definition of the definite integral:
The definite integral of f(x) on the interval a to b is defined as
You might notice that, in this case, n is effectively infinity. Again, while n cannot actually be infinity, we can take the limit, presenting an alternate definition for the definite integral:
In this case these two definitions are identical; it will be helpful to familiarize yourself with both definitions.
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