Problem 9 - Calculus

Question

Use the given values of \(a\) and \(b\) and express the given limit as a definite integral. $$ \lim _{|P| \rightarrow 0} \sum_{i=1}^{n} \frac{\bar{x}_{i}^{2}}{1+\bar{x}_{i}} \Delta x_{i} ; a=-1, b=1 $$

Step-by-Step Solution

Answer

\(\int_{-1}^{1} \frac{x^{2}}{1+x} \, dx\)

Step 1: Identify the Riemann Sum

The given expression \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n} \frac{\bar{x}_{i}^{2}}{1+\bar{x}_{i}} \Delta x_{i}\) is a Riemann sum. Each term in the sum \(\frac{\bar{x}_{i}^{2}}{1+\bar{x}_{i}} \Delta x_{i}\) represents the product of the function \(f(\bar{x}_{i}) = \frac{\bar{x}_{i}^{2}}{1+\bar{x}_{i}}\) and the interval width \(\Delta x_{i}\).

Step 2: Define the Interval

We are given \(a = -1\) and \(b = 1\), which means the interval of integration is \([-1, 1]\). This represents the range over which the sum and ultimately the integral is evaluated.

Step 3: Set up the Definite Integral

In the limit as \(|P| \rightarrow 0\), the Riemann sum becomes a definite integral. Thus, the limit \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n} \frac{\bar{x}_{i}^{2}}{1+\bar{x}_{i}} \Delta x_{i}\) is equivalent to the integral \(\int_{-1}^{1} \frac{x^{2}}{1+x} \, dx\). This is the expression of the limit as a definite integral.

Key Concepts

Riemann Sum

A Riemann Sum is a method used to approximate the total area under a curve on a graph, which is integral. Imagine breaking down the area under a curve into small strips or rectangles. Each of these strips represents a tiny section of the total area you're interested in.
For each rectangle or strip, you will calculate its area by multiplying the height by its width. In the equation from the given problem, the height is represented by the function value \(f(\bar{x}_{i}) = \frac{\bar{x}_{i}^{2}}{1+\bar{x}_{i}}\) and the width is \(\Delta x_{i}\).
The process involves summing up the areas of all these tiny rectangles. This sum \(\sum_{i=1}^{n} \) gives you a close approximation of the curve's total area. As the width of these rectangles (noted as \(|P|\)) gets narrower, this approximation becomes more accurate.

Interval of Integration

The interval of integration defines the specific range over which you calculate the integral. In simple terms, it's the section of the x-axis over which you want to find the area under the curve.
In the context of the exercise, you have been given \(a = -1\) and \(b = 1\). This means you are interested in the interval \([-1, 1]\).
An interval might sometimes be infinite, but in most standard cases, like our example, it is finite. Understanding the interval helps in setting limits for the integral and guides what section of the curve you calculate the integral over.

Limit of a Sum

Taking the limit of a sum is key in transitioning from a Riemann sum to a definite integral. This involves imagining the sum as the number of divisions within the interval becomes infinite and the width of each rectangle approaches zero.
In this scenario, \(|P| \rightarrow 0\) signals that the width of the partitions is decreasing towards zero. This makes the number of partitions, \(n\), increase drastically, creating a sum that perfectly matches the curve.
As this limit process occurs, the sum \(\sum_{i=1}^{n} \) and the function being summed start resembling a continuous function, leading to the integral. Thus, the Riemann Sum becomes the Definite Integral, giving us an exact value for the area under the curve.