Riemann Integration

CDT Addison Bohannon, CDT James Lee, CDT Geoff Phillips

# Introduction

Given a non-negative, real-valued function $f$ on the interval $[a,b]$, the integral of $f$ approaches $S$, the area of the undergraph of $f$. Despite the physical representation of the integral as an area, Bernhard Riemann developed the first rigorous definition of the integral in the middle of the nineteenth century. The integral now has more significance than the anti-operation of the derivative. There are now multiple integrals with increasingly greater range of use, yet Riemann integration is sufficient for nearly all physical problems.

# History

Integration theory found its beginnings in the question of what constitutes a function. Leonhard Euler and Jean D’Alembert’s discussion of the solutions of the wave equation initiated what is now known as Fourier analysis. Daniel Bernoulli, Joseph Lagrange, and Joseph Fourier advanced the mathematical conception of a function and the area which it possesses. The integral returned to an area-based concept as opposed to an anti-derivative concept, at which point, Augustin Cauchy defined the first modern definition of a definite integral as the limit of a sum. His method only allowed integration of certain primitive functions. Peter Gustav Lejeune-Dirichlet advanced his work by focusin on the existence and convergence of Fourier series. Dirichlet’s work almost certainly inspired his contemporary, Bernhard Riemann, who defined the first rigorous area-based definition of the definite integral.

# Riemann Sum

First define a function $f:[a,b]\rightarrow \mathbb{R}$ on an interval $[a,b]$. We then construct a partition pair consisting of two finite sets of points $P,T\subset{[a,b]}$ where $P={x_0,\cdots,x_n}$ and $T={t_1,\cdots,t_n}$. The partition pair must be interlaced such that

(1)
\begin{align} a=x_0\leq{t_1}\leq{x_1}\leq{t_2}\leq{x_2}\leq{\cdots}\leq{t_n}\leq{x_n}=b. \end{align}

Next, we define the intervals of the Riemann sum as

(2)
\begin{align} \Delta{x_i}=x_i-x_{i-1}. \end{align}

We define the mesh of the partition as the length of the largest $\delta{x_i}$. A large value for the mesh is said to be coarse, while a small mesh is said to be fine. With the chosen mesh, we proceed to calculate a Riemann sum,

(3)
\begin{align} R(f,P,T)=\sum^{n}_{i=1}{f(t_i)}\Delta{x_i}}=f(t_i)\Delta{x_i}+f(t_2)\Delta{x_2}+\cdots+f(t_n)\Delta{x_n}, \end{align}

which approximates the area under the graph with the summation of the area of the rectangles.

Figure 1. Graphical representation of the Riemann Sum.

# Riemann Integral

We define the Riemann integral as $I$.

(4)
\begin{align} I=\int^{b}_{a}{f(x)dx=lim_{mesh(P)\rightarrow{0}} R(f,P,T) \end{align}

where $I$ satisfies the following approximation condition:

For all $\epsilon>0$, there exists a $\delta>0$ such that if $P,T$ is any partition pair then mesh $P<\delta\Rightarrow|R-I|<\epsilon$.

# Non-Integrable Functions

It is not safe to assume that a function is integrable without taking a further look at the function.

The following functions are simple functions that upon first glance may appear to be integrable but are not.

(5)
\begin{align} f(x)=\frac{1}{x} \end{align}

on the interval [0,b].

(6)
\begin{align} g(x)=\frac{1}{x^2} \end{align}

on any interval containing 0.

These functions are intrinsically not integrable because the areas Riemann integrable would represent is infinite.

There are other functions that are non-integrable as well. Any function in which the integrand "jumps" around too much is non-integrable. An example of a jump function is the function $h(x)$ where:

(7)
\begin{align} h(x)=\Big\{\begin{matrix}1\ & x\in \mathbb{Q} \\ 0 & x \notin \mathbb{Q} \end{matrix} \end{align}

Thus, the area chosen to represent a single cut in the Riemann sum will be either the width or 0 depending on the selection of $x$. Therefore, for this function, no matter how small the intervals are, you can have a Riemann sum of $0$ or $b-a$.

# Application: Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) relates the two central operations of calculus: differentiation and integration. The FTC is divided into two parts and both parts hold deep significance in calculus. The first part shows that indefinite integration can be reversed by integration.

First Part: If $f:[a,b] \to \mathbb{R}$ is Riemann integrable then its indefinite integral

(8)
\begin{align} F(x)=\int^{x}_{a}f(t)dt, \end{align}

is a continuous function of $x$. The derivative of $F(x)$ exists and equals $f(x)$ at all points $x$ which $f$ is continuous, or

(9)
$$F'(x)=f(x)$$

The second part of the FTC shows that the definite integral of a function can be computed by leveraging any one of the infinite anti-derivatives of a function. This part of the theorem is very significant because it reduced the complication of computing definite integrals.

Second part: Let $f$ be a continuous real-valued function defined on a closed interval $[a,b]$. Let $F$ be an anti-derivative of $f$, that is one of the infinitely many functions such that for all $x$ in $[a,b]$

(10)
$$f(x)=F'(x).$$

Then,

(11)
\begin{align} \int^{b}_{a}f(x)dx=F(b)-F(a). \end{align}

James Gregory first published a statement and proof of a restricted version of the FTC in the 17th century. According to James Stewart , Isaac Newton’s teacher at Cambridge, Isaac Barrow, discovered that the processes of integration and differentiation are inverse ones. The FTC defines the precise relationship between the derivative and the integral. Then, Newton and Leibniz leveraged the concepts into exploitation of the relationship and to develop calculus concepts.

We will now look at a visual proof of the FTC. Observe the picture below.

Figure 1. Visual Proof of the FTC.

We can see from this picture that the FTC works. By definition, the derivative of $A(t)$ is equal to $[A(t + h)-A(t)]/h$ as $h$ tends to zero. Note that the dark blue-shaded region in the illustration is equal to the numerator of the preceding quotient and that the striped region, whose area is equal to its base $h$ times its height $f(t)$, tends to the same value for small $h$. By replacing the numerator, $A(t + h)- A(t)$, by $hf(t)$ and dividing by $h$, $f(t)$ is obtained. Taking the limit as $h$ tends to zero completes the proof of the Fundamental Theorem of Calculus.

There are many important corollaries that follow from the FTC. Two significant ones are listed below. First, the derivative of an indefinite Riemann integral exists almost everywhere and equals the integrand almost everywhere. That is

(12)
\begin{align} f(x)=\frac{d}{dx} \int^{x}_{a}f(t)dt, \end{align}

for all $x$ in $[a,b]$ with $f$ being a real-valued function defined on a closed interval $[a,b]$ and

(13)
$$f(x)=F'(x).$$

Second, every continuous function has an anti-derivative. That is

(14)
\begin{align} F(x)=\int^{x}_{a}f(t)dt+F(a), \end{align}

for all $x$ in $[a,b]$ with $f$ being a real-valued function defined on a closed interval $[a,b]$ and

(15)
$$f(x)=F'(x).$$

# Examples of the FTC

We will now examine several examples of leveraging the FTC to complete problems.

Ex 1.

(16)
\begin{align} \int^{5}_{2}x^2dx \end{align}

We let $f(x)=x^2$ and know that $F(x)=\frac{x^3}{3}$ is the anti-derivative. Therefore,

(17)
\begin{align} \int^{5}_{2}x^2dx=F(5)-F(2)=\frac{125}{3}-\frac{8}{3}=39. \end{align}

Ex 2.

(18)
\begin{align} \frac{d}{dx} \int^{x}_{0}t^3dt \end{align}

Using the first corollary of the FTC above:

(19)
\begin{align} \frac{d}{dx} \int^{x}_{0}t^3dt=f(x)\frac{dx}{dx}-f(0)\frac{d0}{dx}=x^3*1-0^3*0=x^3. \end{align}

# Relevance

Integration Over Unknown Functions

In a situation where it is necessary to find the area under a curve, but the function is unknown, the antiderivative cannot be used to find the integral, or area under the function. However, in this situation, it is possible to use Riemann Integration to find the area under the curve, and therefore the distance the object has traveled. Figure 1 below shows this situation with Riemann Integration applied, with n=8 partitions and a mesh of one half.

Figure 1. Riemann Sums with n=8.

By multiplying the mesh (1/2) by the velocity at the right limit of each partition and then summing these values, we get the Riemann Sum for n=8, which gives us a rough estimate of the area under the curve, or the distance the object has traveled. Because the Riemann Integral is defined as $\lim_{mesh(P)\rightarrow{0}} R(f,P,T)$ where $\R(f,P,T)$ is the Riemann sum, we must find the limit of the Riemann sums as n$\to\infty$. By dividing each partition in half, we decrease the mesh making the sum finer and finer. From a value of 14.06 for the area with n=8, the values quickly decrease to ~13.90 and remain there for n=32 through n=256. At this point, it is clear with have found the limit of the Riemann sums, so we now know that the Riemann Integral of the function depicted in Figure 1 is 13.90. This means the area under the curve is 13.90 and the object has traveled a distance of 13.90 units.

Riemann Integration and Fourier Series

By Stewart's definition, Fourier Series are infinite sums of sines and cosines used to express a given function. Because the series can be expressed as partial sums with n terms, as the number of terms goes up, so does the accuracy in expressing the function. The three images below, taken from Evans M. Harrell III and James V. Herod, illustrate the increasing accuracy of Fourier Series as the number of terms increases.

Figure 1. Fourier Series with One Sine Term.

Figure 2. Fourier Series with Seven Sine Terms.

Here it is easy to see that in much the same way as Riemann Sums, as the number of terms, or n, increases, so does the accuracy of this approximation. In terms of Riemann Integration, Riemann Integration can be used to determine the accuracy of the Fourier Series being used. For instance, using the method for unknown functions shown above, we can calculate the area of the Fourier Series curve and compare that to the area of the actual function's curve. As more terms are added to the partial sum of the Fourier Series, the expression becomes more and more accurate, and the area becomes closer and closer to the area of the function that is being expressed by the Fourier Series. Additionally, for both the Fourier Series and Riemann Sums, as n approaches infinity, the approximations (of area and of the function) approach the actual value.

# References

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