Riemann Sums, Definite Integrals & Fundamental Theorems

1. Partitions and Mesh Size

Let $[a, b] \subset \mathbb{R}$ be a compact interval. A partition $\mathcal{P}$ of $[a, b]$ is a finite ordered sequence of points:

The $k$-th subinterval is $\Delta x_k = x_k - x_{k-1}$. The mesh (or norm) of $\mathcal{P}$ is the maximal subinterval width:

2. Darboux Upper and Lower Sums

For a bounded function $f: [a, b] \to \mathbb{R}$, define the infimum and supremum on each subinterval $[x_{k-1}, x_k]$:

The Lower Darboux Sum $L(f, \mathcal{P})$ and Upper Darboux Sum $U(f, \mathcal{P})$ are:

For any partition $\mathcal{P}$, $L(f, \mathcal{P}) \le U(f, \mathcal{P})$. Furthermore, if $\mathcal{P}^*$ is a refinement of $\mathcal{P}$ ($\mathcal{P} \subseteq \mathcal{P}^*$), adding partition points only increases lower sums and decreases upper sums:

3. The Riemann Integrability Criterion

Define the Lower and Upper Darboux Integrals over all possible partitions $\mathscr{P}$ of $[a, b]$:

Cauchy-Riemann $\varepsilon$-Criterion: A bounded function $f$ is Riemann integrable on $[a, b]$ if and only if for every $\varepsilon > 0$, there exists a partition $\mathcal{P}_\varepsilon$ such that:

Theorem: Every continuous function $f \in C([a, b])$ is Riemann integrable (by Heine-Cantor Uniform Continuity on compact intervals).

1. Tagged Partitions and General Riemann Sums

Choosing an arbitrary evaluation tag $c_k \in [x_{k-1}, x_k]$ inside each subinterval yields the general Riemann sum:

The definite integral is the strict analytical limit as mesh size vanishes:

2. Standard Uniform Partitions ($\Delta x = \frac{b - a}{n}$)

Dividing $[a, b]$ into $n$ equal subintervals with $x_k = a + k \Delta x$ generates classical numerical rules:

• Left Riemann Sum: $c_k = x_{k-1} \implies L_n = \sum_{k=1}^n f(x_{k-1}) \Delta x$
• Right Riemann Sum: $c_k = x_k \implies R_n = \sum_{k=1}^n f(x_k) \Delta x$
• Midpoint Rule: $c_k = \frac{x_{k-1} + x_k}{2} \implies M_n = \sum_{k=1}^n f\left(x_{k-1/2}\right) \Delta x$
• Trapezoidal Rule: The average of Left and Right sums: $$T_n = \frac{L_n + R_n}{2} = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n) \right]$$

3. Asymptotic Error Order

Using Taylor expansions, the truncation error $E(f) = \int_a^b f(x)dx - \text{Approximation}$ scales as:

where $M_1 = \max |f'(x)|$ and $M_2 = \max |f''(x)|$. The Midpoint and Trapezoidal rules achieve second-order convergence $\mathcal{O}(1/n^2)$.

1. Algebraic and Order Properties

For Riemann integrable functions $f, g$ and scalars $\alpha, \beta \in \mathbb{R}$:

1. Linearity: $\int_a^b (\alpha f(x) + \beta g(x)) \, dx = \alpha \int_a^b f(x) \, dx + \beta \int_a^b g(x) \, dx$
2. Subinterval Additivity: For any $c \in (a, b)$, $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$
3. Monotonicity: If $f(x) \le g(x)$ for all $x \in [a, b]$, then $\int_a^b f(x) \, dx \le \int_a^b g(x) \, dx$
4. Integral Triangle Inequality: $$\left| \int_a^b f(x) \, dx \right| \le \int_a^b |f(x)| \, dx$$

2. The Cauchy-Schwarz Inequality for Integrals

For any two real square-integrable functions $f, g \in L^2([a, b])$:

Proof: Consider the quadratic polynomial in $\lambda \in \mathbb{R}$: $P(\lambda) = \int_a^b (\lambda f(x) + g(x))^2 \, dx \ge 0$. Expanding $P(\lambda) = A \lambda^2 + 2B \lambda + C \ge 0$, the discriminant $\Delta = 4(B^2 - AC) \le 0 \implies B^2 \le AC$.

3. The Mean Value Theorem for Definite Integrals

Proof: By the Extreme Value Theorem, $f$ attains minimum $m$ and maximum $M$ on $[a, b]$. Integrating $m \le f(x) \le M$ gives $m(b - a) \le \int_a^b f(x) \, dx \le M(b - a) \implies m \le \frac{1}{b-a}\int_a^b f(x)\,dx \le M$. By Bolzano's Intermediate Value Theorem, $f$ must attain this average value at some point $c \in (a, b)$.

1. The First Fundamental Theorem of Calculus (FTC-1: Differentiation of Accumulation)

Rigorous Proof: Form the Newton difference quotient for $h \ne 0$:

Subtract $f(x) = \frac{1}{h} \int_x^{x+h} f(x) \, dt$ from both sides:

Since $f$ is continuous at $x$, for any $\varepsilon > 0$ there exists $\delta > 0$ such that $|t - x| < \delta \implies |f(t) - f(x)| < \varepsilon$. When $0 < |h| < \delta$, every $t$ in the integration interval satisfies $|t - x| \le |h| < \delta$. Therefore:

Taking the limit as $h \to 0$ proves that $F'(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} = f(x)$. $\blacksquare$

2. The Second Fundamental Theorem of Calculus (FTC-2: Evaluation Theorem)

Proof: From FTC-1, $F(x) = \int_a^x f(t)\,dt$ is an antiderivative of $f$. Since any two antiderivatives differ by a constant on a connected interval, $G(x) = F(x) + C$ for some $C \in \mathbb{R}$.

Evaluating at $x = a$: $G(a) = F(a) + C = \int_a^a f(t)\,dt + C = 0 + C = C$.

Evaluating at $x = b$: $G(b) = F(b) + C = \int_a^b f(t)\,dt + G(a)$.

Subtracting $G(a)$ from both sides establishes: $\int_a^b f(t)\,dt = G(b) - G(a)$. $\blacksquare$

1. Variable Limits and Parameter-Dependent Integrals

Let $f(x, t)$ and its partial derivative $\frac{\partial f}{\partial x}$ be continuous in both variables, and let $u(x), v(x)$ be continuously differentiable functions. Consider the accumulation integral:

2. The Full Leibniz Integral Formula

3. Analytical Derivation via Multi-Variable Chain Rule

Define $H(x, u, v) = \int_u^v f(x, t) \, dt$. The total derivative of $I(x) = H(x, u(x), v(x))$ with respect to $x$ is:

By FTC-1:

Differentiating the integral with respect to parameter $x$ inside the constant limits $[u, v]$ allows passing the derivative under the integral sign: $\frac{\partial H}{\partial x} = \int_u^v \frac{\partial f}{\partial x}(x, t) dt$. Summing these three terms recovers the Leibniz rule.