Problem 16 - Calculus Concepts

Question

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{10}^{\infty}\left(\frac{1}{x}-10\right) d x $$

Step-by-Step Solution

Answer

The improper integral diverges to \(-\infty\).

Step 1: Identify the Improper Integral

The given integral is improper because it involves an infinite limit, specifically from 10 to \( \infty \). We will evaluate this by considering the limit as the upper bound approaches infinity.

Step 2: Rewrite the Integral with a Limit

Rewriting the integral as a limit helps us handle the infinity in the interval. The integral can be rewritten as:\[\lim_{{b \to \infty}} \int_{10}^{b} \left(\frac{1}{x} - 10\right) dx\]

Step 3: Integrate the Function

Integrate \( \left(\frac{1}{x} - 10\right) \) with respect to \( x \). The antiderivative of \( \frac{1}{x} \) is \( \ln |x| \) and the antiderivative of \( -10 \) is \( -10x \). So, the integrated function is:\[ \ln |x| - 10x \]

Step 4: Apply the Limits

Now, apply the limits from 10 to \( b \):\[\left[ \ln |x| - 10x \right]_{10}^{b} = \left( \ln b - 10b \right) - \left( \ln 10 - 100 \right)\]

Step 5: Evaluate the Limit

Take the limit as \( b \to \infty \):\[\lim_{{b \to \infty}} \left( \ln b - 10b \right) + 100 - \ln 10\]Since \( -10b \) approaches \( -\infty \) much faster than \( \ln b \) grows, the expression \( \ln b - 10b \) approaches \( -\infty \).

Step 6: Conclusion on Convergence

The limit is \( -\infty \), indicating that the integral does not converge. Hence, the improper integral diverges.

Key Concepts

Infinite Limit

In the mathematical world, an infinite limit is a significant concept that arises when evaluating integrals that stretch towards infinity. In our example, we look at the integral from 10 to infinity, which creates an infinite limit because it does not have a proper endpoint.

When dealing with integrals, taking the limit as a variable approaches infinity helps us understand the behavior of the integral over an infinite interval. In this specific exercise, we express the integral from 10 to infinity by replacing infinity with a variable, say \( b \), and then examining the behavior as \( b \) tends towards infinity:

• The integral becomes \( \lim_{{b \to \infty}} \int_{10}^{b} \left( \frac{1}{x} - 10 \right) dx \).
• This transformation allows us to apply limit evaluation techniques to examine convergence or divergence.

Antiderivative

The antiderivative, also known as the indefinite integral, is a function that reverses differentiation. Finding the antiderivative is a fundamental step when solving integrals since it provides a function whose derivative results in the original function.

For our integral \( \frac{1}{x} - 10 \), identifying the antiderivative involves finding a function \( F(x) \) such that \( F'(x) = \frac{1}{x} - 10 \). Let's break it down:

• The antiderivative of \( \frac{1}{x} \) is \( \ln |x| \), which comes from the derivative rule \( \frac{d}{dx} \ln |x| = \frac{1}{x} \).
• The antiderivative of the constant \( -10 \) is \( -10x \). This stems from knowing the derivative of \( -10x \) is \(-10\).

Thus, the complete antiderivative is \( \ln |x| - 10x \). This result is used in evaluating the integral's value over the specified limits.

Limit Evaluation

Limit evaluation is crucial in handling improper integrals, particularly those with infinite limits. Once we have the antiderivative, we need to apply the limits to evaluate the integral.

In our task, the process involves substituting the bounds into the antiderivative and finding the limit as the upper bound approaches infinity:

• First, evaluate the antiderivative from 10 to \( b \) where \( b \to \infty \): \[ \left[ \ln |x| - 10x \right]_{10}^{b} \]
• This results in \( \ln b - 10b - (\ln 10 - 100) \).
• Within this expression, evaluate the limit as \( b \) approaches infinity: - Because \( -10b \) tends to \(-\infty\) way faster than \( \ln b \) can increase, \( \ln b - 10b \) leads to \(-\infty\).

Thus, determining how this limit behaves tells us whether the original integral converges or diverges.

Divergent Integral

An integral is classified as divergent if, after evaluation, it does not settle to a specific finite value. This often implies that the integral's area under the curve stretches to infinitely large values.

In the context of this exercise, we learn that:

• After thoroughly evaluating the integral using its antiderivatives and limits, the expression approaches \(-\infty\).
• Since the limit of this integral as it stretches to infinity results in \(-\infty\), it fails to converge to a constant value.

Therefore, we conclude the integral is divergent, as the calculations show it does not stabilize, affirming its nonexistence as a finite number. It's essential to recognize divergent integrals as they depict functions whose evaluated areas are unbounded.