Problem 21 - Applied Calculus

Question

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{2}^{\infty} \frac{1}{x} d x $$

Step-by-Step Solution

Answer

The integral is divergent.

Step 1: Identify the Type of Integral

The given integral is \( \int_{2}^{\infty} \frac{1}{x} \, dx \). It is an improper integral because it has an infinite limit of integration.

Step 2: Set Up the Limit

To evaluate the improper integral, we rewrite it as a limit: \[ \lim_{b \to \infty} \int_{2}^{b} \frac{1}{x} \, dx. \]

Step 3: Compute the Definite Integral

Find the antiderivative of \( \frac{1}{x} \), which is \( \ln |x| + C \). Now compute the definite integral from 2 to \( b \): \( \int_{2}^{b} \frac{1}{x} \, dx = \ln |b| - \ln |2| \).

Step 4: Evaluate the Limit

Apply the limit to the result from Step 3: \[ \lim_{b \to \infty} (\ln |b| - \ln |2|). \] As \( b \to \infty \), \( \ln |b| \to \infty \). Therefore, \( \lim_{b \to \infty} (\ln |b| - \ln |2|) = \infty - \ln 2 = \infty \).

Step 5: Conclude about Convergence or Divergence

Since the limit evaluated in Step 4 is \( \infty \), the integral \( \int_{2}^{\infty} \frac{1}{x} \, dx \) is divergent.

Key Concepts

Convergence

Convergence is a critical concept when dealing with improper integrals in calculus. An improper integral is said to be convergent if it evaluates to a finite number.
This means that as we extend the limit of integration towards infinity, the area under the curve approaches a specific, finite value.
However, not all improper integrals are convergent.

• A common technique to test for convergence is to set up the integral as a limit problem. This involves applying a limit process to the upper or lower end of the interval where the integral is improper.
• If the result of this limit is a finite number, the integral is convergent.
• Convergence is important because it determines whether the improper integral has a meaningful value that represents a total accumulated quantity.

In the provided exercise, we found that the integral did not converge, as the limit approached infinity.

Divergence

Divergence refers to the behavior of an improper integral that does not evaluate to a finite number. When an integral is divergent, it means the area under the curve grows without bound as you extend the interval.
Essentially, this means there is no total finite quantity; the integral has no meaningful numerical value.
Divergence is typically indicated by a limit approaching infinity or negative infinity.

• Analyzing divergence helps identify cases where a mathematical description as an improper integral does not compute to a useful real-world quantity.
• In the context of the exercise, the integral from 2 to infinity of \( \frac{1}{x} \, dx \) was found to be divergent because the limit yielded infinity.
• This evaluation shows the area under the curve does not settle to a fixed size but extends indefinitely.

Recognizing divergence helps in understanding the limitations of integral definitions and applications.

Limits in Calculus

Limits are foundational to the concept of improper integrals. They are essential for assessing convergence and divergence. When dealing with improper integrals, limits allow one to handle infinity in calculus, which is crucial because direct computation with infinity isn't feasible.
Limits provide a way to ascertain whether the area accumulation continues to expand indefinitely or settles to a finite value.
In solving improper integrals, setting up the integral with a limit is often the first step.

• For example, replacing the upper bound of an integral with a variable like \( b \), and then considering the behavior as \( b \to \infty\), allows for the calculation of areas extending to infinity.
• If this limit results in a finite number, the original integral is convergent; if infinite, it is divergent.
• Limits enable dealing with complex calculus problems that involve infinity, making them an essential tool in mathematical analysis.

Understanding how limits work helps demystify the behaviors of functions at their extremes and is a key skill in advanced calculus.