1: Review of Riemann sums and definite integrals

Section 1: Integration Techniques - 1: Review of Riemann Sums and Definite Integrals

This section revisits the foundational concepts that define the definite integral: Riemann sums and their limit. Understanding this is crucial before advancing to sophisticated integration techniques.

The Area Problem & Riemann Sums
Imagine finding the area under the curve $y = f(x)$ between $x = a$ and $x = b$, where $f(x)$ is continuous and non-negative. A Riemann sum approximates this area. Here's how:

1. Partition: Divide $[a, b]$ into $n$ subintervals using points: $a = x_0 < x_1 < x_2 < ... < x_n = b$.
2. Widths: The width of the $i^\text{th}$ subinterval $[x_{i-1}, x_i]$ is $\Delta x_i = x_i - x_{i-1}$. Often, regular partitions ($\Delta x_i = \Delta x = \frac{b-a}{n}$) are used.
3. Sample Points: Choose a sample point $x_i^*$ within each subinterval $[x_{i-1}, x_i]$ (e.g., left endpoint, right endpoint, midpoint).
4. Sum: Form the sum of areas of rectangles built on each subinterval: $$R_n = \sum_{i=1}^{n} f(x_i^*) \Delta x_i$$ This sum $R_n$ is a Riemann sum. It represents the total area of the rectangles approximating the region under the curve.

The Definite Integral as a Limit
The approximation improves as the number of subintervals $n$ increases and the maximum subinterval width ($\|\Delta x\| = \max \Delta x_i$) approaches zero. The definite integral of $f(x)$ from $a$ to $b$ is defined as the limit of these Riemann sums:

$$\int_{a}^{b} f(x) dx = \lim_{\|\Delta x\| \to 0} \sum_{i=1}^{n} f(x_i^*) \Delta x_i$$

If this limit exists (which it does for continuous functions on $[a, b]$), $f$ is said to be integrable on $[a, b]$. Geometrically, for non-negative $f(x)$, this integral represents the exact net area between the curve and the x-axis over $[a, b]$.

Key Properties of Definite Integrals
Definite integrals obey essential algebraic rules vital for computation and manipulation:

• Linearity: $$\int_{a}^{b} [c_1 f(x) + c_2 g(x)] dx = c_1 \int_{a}^{b} f(x) dx + c_2 \int_{a}^{b} g(x) dx$$
• Additivity over Intervals: For $a < c < b$, $$\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$$
• Order Preservation: If $f(x) \leq g(x)$ on $[a, b]$, then $$\int_{a}^{b} f(x) dx \leq \int_{a}^{b} g(x) dx$$
• Reversing Limits: $\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx$.
• Integral of a Constant: $\int_{a}^{b} c dx = c(b - a)$.

While the Riemann sum definition provides the core meaning, the Fundamental Theorem of Calculus (covered next) furnishes the primary tool for actually evaluating definite integrals. Mastery of this limit process and integral properties is essential for understanding advanced integration methods.