Problem 20 - Calculus: Early Transcendentals

Question

Express the limit as a definite integral on the given interval. \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \frac{x_i^*}{(x_i^*)^2 + 4} \, \Delta x \), [1, 3]

Step-by-Step Solution

Answer

The limit is expressed as the integral \( \int_{1}^{3} \frac{x}{x^2+4} \; dx \).

Step 1: Understand the Summation

The given expression is a limit of a summation, which is often an indicator of a Riemann sum. A Riemann sum is used to approximate the integral of a function. The expression inside the summation is the function evaluated at some point in the subinterval, and \( \Delta x \) represents the change in \( x \) or the width of subintervals.

Step 2: Identify the Function and Interval

The function here is \( \frac{x^*}{(x^*)^2 + 4} \), where \( x^* \) is a specific point in the \( i^{th} \) subinterval, typically of the form \( a + i\Delta x \) or \( a + (i+1)\Delta x \). The interval mentioned is \([1, 3]\), meaning \( a = 1 \) and \( b = 3 \).

Step 3: Determine \( \Delta x \) and Limits of Integration

\( \Delta x \) is found using the formula \( \Delta x = \frac{b-a}{n} = \frac{3-1}{n} = \frac{2}{n} \). As \( n \to \infty \), \( \Delta x \to 0 \).

Step 4: Set Up Riemann Sum as an Integral

The Riemann sum, \( \sum_{i=1}^n \frac{x_i^*}{(x_i^*)^2 + 4} \Delta x \), becomes an integral: \[ \int_{1}^{3} \frac{x}{x^2+4} \; dx \] This integral results from the function within the summation.

Key Concepts

Understanding Riemann Sums

Riemann sums are a fundamental concept used to approximate the area under a curve or the integral of a function. When you break down an interval into small subintervals, each of these subintervals has a corresponding width, noted as \( \Delta x \). By choosing a point within each subinterval, typically noted as \( x_i^* \), we evaluate the function at these points. Putting it all together, a Riemann sum looks like this:

• \(\sum_{i=1}^n f(x_i^*) \Delta x\) - where \( f(x_i^*) \) is the function evaluated at the chosen points.

The more split intervals we have, the closer this sum approximates the actual area under the curve. This principle is key when transitioning from a Riemann sum to a definite integral as the number of intervals, \( n \), approaches infinity.

Exploring the Limit

When we discuss limits in the context of Riemann sums, we're talking about the behavior of a function as \( n \) approaches infinity. In simpler terms, it's examining what happens when you make the subintervals infinitely small:

• The width of each interval, \( \Delta x \), gets smaller and smaller.
• The sum \( \sum_{i=1}^n f(x_i^*) \Delta x \) becomes a more accurate representation of the integral \( \int f(x) \, dx \).

Thus, in the context of definite integrals, taking the limit as \( n \to \infty \) transforms a Riemann sum into a precise account of the total area under the curve defined by a function on an interval. In our example, as \( n \to \infty \), the Riemann sum \( \sum_{i=1}^n \frac{x_i^*}{(x_i^*)^2 + 4} \Delta x \) transitions into the integral \( \int_{1}^{3} \frac{x}{x^2+4} \; dx \).

Integral Approximation with Limits

Integral approximation involves estimating the value of a definite integral. Through Riemann sums, we see how subintervals and sampled points can approximate the area under a curve. The better the approximation desired, the more subintervals are needed:

• Each subinterval gives a small rectangle contributing to the area estimation.
• Using limits, we refine those rectangles further for greater accuracy.

In practical terms, approximation often involves computational tools to handle many subintervals efficiently. For the integral example we're working on, understanding that \( \int_{1}^{3} \frac{x}{x^2 + 4} \; dx \) originates from an approximation process helps deepen the comprehension of calculus techniques.

Evaluating Functions Accurately

The process of function evaluation in the context of integrals involves understanding how the function behaves over a specific interval by sampling points:

• For each subinterval, select a value \( x_i^* \); this is where the function \( f(x) = \frac{x}{x^2 + 4} \) is evaluated in our example.
• The choice of \( x_i^* \) and accuracy in computing \( f(x_i^*) \) are crucial to correct approximation.

Consistently accurate function evaluation ensures that the approximation accurately captures the function's curve, leading to a precise integration outcome. This understanding bridges discrete approximations and continuous integral calculations.