Problem 33 - Calculus

Question

Show that \(\int_{a}^{b} x d x=\frac{1}{2}\left(b^{2}-a^{2}\right)\) by completing the following argument. For the partition \(a=x_{0}

Step-by-Step Solution

Answer

The integral is \(\int_{a}^{b} x \, dx = \frac{1}{2}(b^2 - a^2)\), using Riemann sum formulation and telescopic cancellation.

Step 1: Understand the Riemann Sum Concept

We start with a simple partition of the interval \([a, b]\) represented as \(a = x_0 < x_1 < \cdots < x_n = b\). The choice for \( \bar{x}_i\) is important because it represents a sample point for the partition such that \( \bar{x}_i = \frac{1}{2}(x_{i-1} + x_i)\).

Step 2: Formulate the Riemann Sum

Using the formula for \(R_P\), we have: \[ R_P = \sum_{i=1}^{n} \bar{x}_{i} \Delta x_{i} = \sum_{i=1}^{n} \frac{1}{2}(x_{i-1} + x_i)(x_i - x_{i-1}) \] This sum defines a Riemann sum for the function \(f(x) = x\) over the partition with sample points \(\bar{x}_i\).

Step 3: Expand and Simplify the Sum

Next, expand the Riemann sum: \[ R_P = \frac{1}{2} \sum_{i=1}^{n} (x_i^2 - x_{i-1}^2) \]This uses the identity \((a + b)(x_i - x_{i-1}) = x_i^2 - x_{i-1}^2\).

Step 4: Use the Telescoping Sum Structure

The sum becomes a telescoping sum: \[ \frac{1}{2} ((x_n^2 - x_{n-1}^2) + (x_{n-1}^2 - x_{n-2}^2) + \cdots + (x_1^2 - x_0^2)) \]Most terms cancel, leaving: \[ \frac{1}{2} (x_n^2 - x_0^2) \] Equating this to \(\frac{1}{2}(b^2 - a^2)\) since \(x_n = b\) and \(x_0 = a\).

Step 5: Conclude With the Limit Definition of the Integral

Finally, taking the limit as the partition becomes finer (\( n \rightarrow \infty \)), the Riemann sum \(R_P\) approaches the definite integral: \[ \lim_{n \to \infty} R_P = \int_{a}^{b} x \, dx = \frac{1}{2}(b^2 - a^2) \] Thus, the integration is shown based on partition simplification and the limit process.

Key Concepts

Riemann Sum

The Riemann sum is a fundamental concept used to understand the definite integral. It's essentially a way to approximate the area under a curve, working as a precursor to integration. Here's how it unfolds.

• Think of breaking the area under a curve between two intervals, \[a, b\], into narrow rectangles called partitions.
• The choice of where to sample the function value within each partition is crucial. In our case, the sample point is \( \bar{x}_i = \frac{1}{2}(x_{i-1} + x_i) \), representing the midpoint of each partition segment.

In this exercise, our goal is to express \([a, b]\) as a sum of areas under small intervals and show what they approximate. This transformation from a sum to a limit allows us to explore the nuances of integration, serving as an introductory step toward evaluating the definite integral.

Telescoping Sum

A telescoping sum is a clever mathematical technique that simplifies a complex sum into something more manageable. It does so by eliminating most intermediate elements, making it easier to calculate.

• Expand the sum as shown: \R_P = \frac{1}{2} \sum_{i=1}^{n} (x_i^2 - x_{i-1}^2)\.
• Notice how terms like \(x_{n-1}^2 - x_{n-2}^2\) cancel successively, leaving only the first and last terms.

This leaves us with just \((x_n^2 - x_0^2)\), simplifying the calculation significantly. It's akin to peeling layers off an onion until only the core remains—once the overlapping terms cancel, it's much simpler to solve or approximate the initial sum.

Integration Limits

Integration limits specify the endpoints \([a, b]\) of the definite integral. These bounds dictate where the area under a curve starts and ends, functioning as the key parameters for calculating the integral.

• The limits directly influence the partition points \(x_0\) and \(x_n\), corresponding to points \(a\) and \(b\), respectively.
• In our solution, the expression \(\frac{1}{2}(b^2 - a^2)\) ultimately embodies the sorted integration of the function \(f(x) = x\) over that interval.

When taking limits as the partition becomes finer (more subintervals), the Riemann sum turns into the exact integral we are computing. Thus, the integration limits help finalize the transition from the approximated sum into the precise area under the curve, fulfilling the description of a definite integral.