Integration Theory

While introductory calculus treats integration as “the reverse of differentiation,” Analysis treats it as a foundational process of accumulation. We begin with a rigorous definition of the Riemann Integral using the framework of Darboux sums.

The Riemann-Darboux Integral

Let $f:[a,b]\to \mathbb{R}$ be a bounded function. We define a partition $P$ of $[a,b]$ as a finite set of points $\{x_0,x_1,\dots ,x_n\}$ such that $a=x_0<x_1<\cdots <x_n=b$.

Lower and Upper Sums

• Let $m_i=\inf \{f(x):x\in [x_{i-1},x_i]\}$
• Let $M_i=\sup \{f(x):x\in [x_{i-1},x_i]\}$

The Lower Sum $L(f,P)$ and Upper Sum $U(f,P)$ are: $L(f,P)={\sum }_{i=1}^n{m}_i\Delta x_i\text{and}U(f,P)={\sum }_{i=1}^n{M}_i\Delta x_i$

The Integrability Criterion

We define the Lower Integral $\underline{{\int }_a^b}f$ as the supremum of all lower sums, and the Upper Integral $\overline{{\int }_a^b}f$ as the infimum of all upper sums. A function $f$ is Riemann Integrable if: $\underline{{\int }_a^b}f=\overline{{\int }_a^b}f$ The common value is denoted ${\int }_a^{b}f(x)dx$.

Important Integrability Results

Not every function is integrable. For example, the Dirichlet function (which is 1 on rationals and 0 on irrationals) is not Riemann integrable because any interval contains both rationals and irrationals, making $L(f,P)=0$ and $U(f,P)=b-a$.

Theorem: Sufficient Conditions

1. Every continuous function on $[a,b]$ is Riemann integrable.
2. Every monotonic function on $[a,b]$ is Riemann integrable.
3. Every bounded function with only finitely many points of discontinuity is Riemann integrable.

Properties of the Integral

The Riemann integral satisfies several fundamental algebraic properties:

• Linearity: $\int (af+bg)=a\int f+b\int g$.
• Additivity: ${\int }_a^{b}f+{\int }_b^{c}f={\int }_a^{c}f$.
• Monotonicity: If $f\le g$ on $[a,b]$, then ${\int }_a^{b}f\le {\int }_a^{b}g$.

Mean Value Theorem for Integrals

If $f$ is continuous on $[a,b]$, there exists a point $c\in [a,b]$ such that: $f(c)=\frac{1}{b-a}{\int }_a^{b}f(x)dx$ This value $f(c)$ is the average value of the function over the interval.

Generalizations: Lebesgue Integration

Riemann integration has limitations, particularly when dealing with sequences of functions. To rectify this, modern analysis uses Lebesgue Integration, which partitions the range (y-axis) of the function rather than the domain (x-axis). This allows for the integration of much “wilder” functions and provides a more robust framework for probability theory and functional analysis.

Python: Computing Riemann Sums

We can visualize the convergence of the lower and upper sums using a simple Python script.

def riemann_sums(f, a, b, n):
    dx = (b - a) / n
    nodes = [a + i * dx for i in range(n + 1)]
    
    lower_sum = 0
    upper_sum = 0
    
    for i in range(n):
        # Sample function at many points to approximate inf/sup
        sample_x = [nodes[i] + j * (dx / 100) for j in range(101)]
        sample_y = [f(x) for x in sample_x]
        
        lower_sum += min(sample_y) * dx
        upper_sum += max(sample_y) * dx
        
    return lower_sum, upper_sum

# Test with f(x) = x^2 on [0, 1]
f = lambda x: x**2
for n in [10, 100, 1000]:
    L, U = riemann_sums(f, 0, 1, n)
    print(f"n={n}: Lower={L:.4f}, Upper={U:.4f}, Diff={U-L:.4f}")

Summary

Integration theory is the science of accumulation. By formalizing the integral using Darboux sums, we create a tool that is not only useful for finding areas but also for defining new types of functions and understanding the convergence of signals and series. It is the precursor to measure theory and the foundation of modern physical laws.

Exercise