The Riemann Integral: Definition and Existence

The Area Problem

How can one find the area under the curve $y = f(x)$ from $a$ to $b$? If $f = \text{const}$, the answer is trivial. If $f$ is a polynomial, one can divide into trapezoids. But what about an arbitrary function?

Riemann's idea (1854): divide the segment $[a, b]$ into small parts, on each part approximately replace the function with a constant, sum up the rectangles.

Riemann Sums

A partition $T$: $a = x_0 < x_1 < \ldots < x_n = b$. For each subinterval $[x_{i-1}, x_i]$ of length $\Delta x_i$ choose a point $\xi_i \in [x_{i-1}, x_i]$.

Riemann sum: $S(f, T, \xi) = \sum_i f(\xi_i)\Delta x_i$.

Diameter of the partition: $\lambda = \max \Delta x_i$.

A function $f$ is Riemann integrable on $[a, b]$ if the limit of Riemann sums exists as $\lambda \to 0$, independent of partition and points $\xi_i$. This limit is the definite integral $\int_a^b f(x)dx$.

Riemann's Criterion

Introduce the upper and lower Darboux sums: $U(f, T) = \sum M_i, \Delta x_i$ and $L(f, T) = \sum m_i, \Delta x_i$, where $M_i = \sup f$ and $m_i = \inf f$ on $[x_{i-1}, x_i]$.

$f$ is integrable $\iff$ for any $\varepsilon > 0$ there exists a partition $T$ such that $U(f,T) - L(f,T) < \varepsilon$.

Theorem: A function continuous on $[a, b]$ is integrable. A monotonic function is integrable. A bounded function with a finite number of discontinuities is integrable.

Properties of the Integral

$\int_a^a f, dx = 0$, $\int_a^b f, dx = -\int_b^a f, dx$.

Linearity: $\int(\alpha f + \beta g) = \alpha \int f + \beta \int g$.

Additivity: $\int_a^b f = \int_a^c f + \int_c^b f$.

Estimate: if $m \leq f(x) \leq M$ on $[a, b]$, then $m(b-a) \leq \int_a^b f, dx \leq M(b-a)$.

Mean value theorem: $\int_a^b f, dx = f(c)(b-a)$ for some $c \in [a, b]$.

Improper Integrals

The integral $\int_1^\infty \frac{dx}{x^p}$ converges for $p > 1$ and diverges for $p \leq 1$. This is the standard comparison test: if the function decreases "sufficiently rapidly", the integral converges.

Gaussian integral: $\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}$. This is a remarkable result, used in probability theory.

The Connection between the Integral and Probability

The Riemann integral is the language of probability theory. If a continuous random variable $X$ has density $f(x) \geq 0$, then $P(a \leq X \leq b) = \int_a^b f(x), dx$. The normalization condition $\int_{-\infty}^\infty f(x), dx = 1$ is precisely an improper integral. The normal distribution $N(\mu, \sigma^2)$ with density $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ is normalized thanks to the Gaussian integral.

Mathematical expectation $E[X] = \int x f(x), dx$, variance $\mathrm{Var}[X] = \int (x-\mu)^2 f(x), dx$—all these are integrals. Probability theory is literally an applied theory of the integral.

Lebesgue's Criterion for Riemann Integrability

A function is bounded on $[a, b]$ and Riemann integrable if and only if the set of its points of discontinuity has measure zero. This means that a function may have infinitely many points of discontinuity—as long as they do not "occupy" any interval. The Dirichlet function (1 on rationals, 0 on irrationals) is nowhere continuous and not integrable in the Riemann sense, but is integrable in a more general sense (in the Lebesgue sense).

The Integral as the Limit of Integral Sums

Practical significance: numerical integration. Rectangle method: $\int_a^b f(x) dx \approx h \cdot \sum f(x_i)$. Trapezoid method: $\int_a^b f dx \approx h \cdot \left (f(a)/2 + f(x_1) + \ldots + f(x_{n-1}) + f(b)/2 \right )$. Simpson's method: parabolic rule, error $O(h^4)$. In modern packages (SciPy, MATLAB) adaptive schemes are used, automatically refining the partition where the function varies rapidly.

Lebesgue's Criterion for Riemann Integrability

A function is Riemann integrable if and only if the set of its discontinuities has Lebesgue measure zero (Lebesgue's theorem, 1901). This is the precise boundary: the Dirichlet function (1 on rationals, 0 on irrationals) is not Riemann integrable, as it is discontinuous everywhere. The Thomae function (0 on irrationals, $1/q$ at $p/q$) is integrable: its discontinuity points are the rational numbers, which are countable, and countable sets have measure zero. This criterion explains why "most" bounded functions are integrable—their discontinuities form "small" sets.

Thought question: The function $f(x) = \sin(1/x)$ for $x \in (0, 1]$ oscillates without bound near zero. Is it Riemann integrable on $[0, 1]$ (if we define $f(0) = 0$)?

Darboux Sums and Precise Characterization of Integrability

Lower Darboux sum: $L(P, f) = \sum m_i \Delta x_i$, where $m_i = \inf f$ on the subinterval. Upper sum: $U(P, f) = \sum M_i \Delta x_i$, where $M_i = \sup f$. A function is Riemann integrable if and only if $\inf_P U(P, f) = \sup_P L(P, f)$ (Darboux's definition)—this is equivalent to the original definition via integral sums. Geometric meaning: the integral exists when the upper and lower stepwise approximations "merge". For monotonic functions this is always fulfilled: $U - L \leq (f(b) - f(a)) \cdot \max \Delta x_i \to 0$ as the partition is refined. For the Dirichlet function ($1$ on $\mathbb{Q}$, $0$ on $\mathbb{R} \setminus \mathbb{Q}$): $L = 0$, $U = 1$ for any partition—the integral does not exist.