Introduction to the Riemann Integral

Jeremy Knight

(This page uses Math Player plugin to view formulas)


Bernhard Riemann (1826-1866) was a German mathematician who was credited with some of the most important developments in modern mathematics. One operation that bears his name is the Riemann Integral.

What is an Integral?

An integral is the mathematical term for the area associated with a function. There are different types of integrals, but the simplest type of integral is called the Riemann integral. The Reimann integral represents the area between the graph of a function and the x-axis. So, we will investigate some strategies for finding this area.

The Riemann integral in 2-dimensions is defined on a given interval of the x. With symbols, we write the integral of amath f(x) endamath over x from a to b as:

amath \int_a^b f(x) dx endamath

Here the f(x) is the function and the dx tells us that we are finding the area between the function and the x-axis.

Methods and Strategies

We want to approximate the integral of a function using the area of familiar shapes. To do this we must first divide the interval into a Partition, which just means that we want to split up the distance between a and b into equally spaced subintervals.

Example 1:
Approximate amath \int_0^10 x^2 dx endamath. That is, we need to find the area underneath the parabola amath y=x^2 endamath beween the x-axis and the graph which is the shaded region in the graph to the right.

Step 1: First we will divide the interval [0,10] into the partition P={0,2,4,6,8,10}. This splits the x-axis into the sub-intervals


Step 2: Now we will calculate the value for amath f(x)=x^2 endamath for each of the interval points from step 1.

amath f(0)=0 , f(2)=4 , f(4)=16 , f(6)=36 , f(8)=64 , f(10)=100

Integral of f(x)=x^2 on [0,10]
Step 3:We will now approximate the area in one of three ways:

Try It:

Now it is your turn. Use the trapezoid method to approximate the value of amath \int_0^10 x^2 dx endamath using a partition of 10 subintervals.

In other words, we will divide the interval [0,10] into the partition P={0,1,2,3,4,5,6,7,8,9,10}. When we plot the trapezoids, we will get the graph to the right.

Now download the RiemannSums.m file and open it up in Matlab.

Use this program to test your approximation for this integral.

Programming Exploration

Let's use MatLab to explore these methods for finding an approximation for an integral. Use the RiemannSums.m to approximate the following integrals to within an error of 10-3 by refining the number of intervals until the error is small enough.

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