gen-define-the-riemann-integral-and-give-one-short-worked-exampl-767f

Define the Riemann integral and give one short worked example.

# Riemann Integral Definition Let f be a bounded function on [a,b]. A **partition** P of [a,b] is a finite set of points: $$P = \{x_0, x_1, ..., x_n\} \text{ where } a = x_0 < x_1 < ... < x_n = b$$ For each subinterval [x_{i-1}, x_i], let: - M_i = sup{f(x) : x ∈ [x_{i-1}, x_i]} - m_i = inf{f(x) : x ∈ [x_{i-1}, x_i]} The **upper and lower Riemann sums** are: $$U(f,P) = \sum_{i=1}^{n} M_i(x_i - x_{i-1})$$ $$L(f,P) = \sum_{i=1}^{n} m_i(x_i - x_{i-1})$$ The function f is **Riemann integrable** on [a,b] if: $$\sup_P L(f,P) = \inf_P U(f,P)$$ This common value is the **Riemann integral**: $\int_a^b f(x)\,dx$ --- ## Worked Example Find $\int_0^1 x^2\,dx$ using the partition with n equal subintervals. **Solution:** Let $x_i = \frac{i}{n}$ for i = 0,1,...,n. Since f(x) = x² is increasing on [0,1]: - m_i = f(x_{i-1}) = (i-1)²/n² - M_i = f(x_i) = i²/n² $$L(f,P) = \sum_{i=1}^{n} \frac{(i-1)^2}{n^2} \cdot \frac{1}{n} = \frac{1}{n^3}\sum_{i=0}^{n-1} i^2 = \frac{1}{n^3} \cdot \frac{(n-1)n(2n-1)}{6} \to \frac{1}{3}$$ Similarly, $U(f,P) \to \frac{1}{3}$ as n → ∞. Therefore: $\boxed{\int_0^1 x^2\,dx = \frac{1}{3}}$