Problem 74 - Calculus: Early Transcendentals

Question

Express the limit as a definite integral. \( \displaystyle \lim_{n \to \infty} \frac{1}{n} \sum^{n}_{i = 1} \frac{1}{1 + (i/n)^2} \)

Step-by-Step Solution

Answer

The limit is expressed as \( \int_{0}^{1} \frac{1}{1 + x^2} \, dx \).

Step 1: Identify the Riemann Sum

Recognize that the limit expression given is a Riemann Sum for a definite integral. The expression \( \frac{1}{n} \sum_{i=1}^{n} f(x_i) \) approximates the integral of a function \( f(x) \) over an interval \([a, b]\). Here, the function \( f(x) = \frac{1}{1 + x^2} \) is evaluated at points \( x_i = \frac{i}{n} \).

Step 2: Determine the Interval

Since \( x_i = \frac{i}{n} \) and \( i \) ranges from \( 1 \) to \( n \), the interval for \( x \) is \([0, 1]\). This is because the smallest and largest values of \( x_i \) are \( \frac{1}{n} \) and \( 1 \), respectively, as \( n \to \infty \).

Step 3: Write the Corresponding Definite Integral

The Riemann Sum \( \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1+(\frac{i}{n})^2} \) corresponds to the definite integral of the function \( \frac{1}{1 + x^2} \) over the interval \([0, 1]\). Thus, the integral is \[ \int_{0}^{1} \frac{1}{1 + x^2} \, dx \].

Key Concepts

Riemann Sum

A Riemann Sum is a mathematical tool used to approximate the area under a curve. Essentially, it breaks down an interval into smaller subintervals and calculates the sum of the areas of rectangles that approximate the curve over these subintervals. This method is foundational to understanding integration.- A given interval \([a, b]\) is divided into \(n\) subintervals of equal width \(\Delta x = \frac{b-a}{n}\).- On each subinterval, a sample point \(x_i\) is selected, and the function \(f(x)\) is evaluated at these points.- Multiply the function value by \(\Delta x\) for each subinterval and sum up all these values.As the number of subintervals \(n\) increases, the approximation becomes better. In the limit as \(n \rightarrow \infty\), the Riemann Sum converges to the definite integral of the function over \([a, b]\). In the exercise, the sum \(\frac{1}{n} \sum^{n}_{i = 1} \frac{1}{1 + (i/n)^2}\) is an example of a Riemann Sum.

Limit of a Sum

The limit of a sum involves taking the limit of a Riemann Sum as the number of subintervals grows infinitely large, \(n \rightarrow \infty\). This process transforms the Riemann Sum into a definite integral, providing an exact value rather than an approximation.- The integral is equivalent to the sum when \(n\) is infinite.- This limit represents the exact area under the curve of \(f(x)\) over the defined interval.- In the given problem, as \(n\) increases, the sum \(\frac{1}{n} \sum^{n}_{i = 1} \frac{1}{1 + (i/n)^2}\) approaches the exact value of the integral \[\int_{0}^{1} \frac{1}{1 + x^2} \, dx\].This method transforms an approximation into a precise calculation, underlining the importance of understanding the connection between Riemann Sums and definite integrals.

Interval Identification

Identifying the correct interval over which the function is integrated is crucial to solving a definite integral. The interval determines the bounds of integration.- For a proper identification, determine the smallest and largest values of the sample points \(x_i\), which, for \(x_i = \frac{i}{n}\), range from \(\frac{1}{n}\) to \(1\) as \(i\) goes from \(1\) to \(n\).- When \(n\) tends to infinity, the smallest value \(\frac{1}{n}\) approaches zero.Thus, the interval identified is \[0, 1\], which indicates that the function \(\frac{1}{1 + x^2}\) is integrated from zero to one.

Function Integration

Function integration is the process of finding the integral of a function over a specified interval. It provides the total accumulated value, which can be seen as the area under the curve of the function within that interval.- In the exercise, the integration of \(\frac{1}{1 + x^2}\) over the interval \[0, 1\] results in a definite integral.- The definite integral is represented as \[\int_{0}^{1} \frac{1}{1 + x^2} \, dx\].- By evaluating this integral, one can determine the exact region under the curve from \(x = 0\) to \(x = 1\).Understanding this integration process is essential for solving any problem involving definite integrals. It sums up the continuous contributions of the function along the specified range.