Problem 15 - Calculus Concepts

Question

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

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Identify the Improper Nature

The integral \( \int_{1}^{\infty} \frac{10}{x} \, dx \) is improper because the upper limit of integration is infinity. To evaluate it, we'll consider it as a limit problem.

Step 2: Set Up the Integral as a Limit

Rewrite the improper integral by replacing the infinity with a variable, say \( b \), and then take the limit as \( b \) approaches infinity: \[ \int_{1}^{b} \frac{10}{x} \, dx. \]

Step 3: Integrate the Function

Find the antiderivative of \( \frac{10}{x} \). The antiderivative is \( 10 \ln |x| \). So, evaluate the integral from \(1\) to \(b\): \[ \left. 10 \ln |x| \right|_1^b = 10 \ln |b| - 10 \ln 1. \]

Step 4: Simplify and Evaluate the Limit

Since \( \ln 1 = 0 \), the expression simplifies to \( 10 \ln |b| \). Now evaluate the limit: \[ \lim_{b \to \infty} 10 \ln |b|. \] As \( b \to \infty \), \( 10 \ln |b| \to \infty \).

Step 5: Conclusion of Divergence

Since \( \lim_{b \to \infty} 10 \ln |b| = \infty \), the original improper integral diverges.

Key Concepts

Infinity in Calculus

In calculus, the concept of infinity is crucial when dealing with limits and improper integrals. Whenever you see the symbol \( \infty \), it represents a value that grows without bound. It is not an actual number, but rather a way to describe behavior that exceeds any finite measurement. For example, when you calculate an integral with an upper limit of infinity, like \( \int_{1}^{\infty} \frac{10}{x} \, dx \), it means you're summing over all possible values starting from 1 to an unbounded end. Understanding infinity helps mathematicians and students analyze how functions behave as they "approach" infinitely large values. This is fundamental in calculus because it allows us to extend our study beyond finite numbers, enabling a comprehensive examination of growth, limits, and convergence.

Limit Evaluation

Limit evaluation is a vital technique in calculus used to solve improper integrals. When dealing with an improper integral like \( \int_{1}^{\infty} \frac{10}{x} \, dx \), the infinite limit indicates that you cannot compute it directly. Instead, you analyze it as a limit problem. This involves replacing the infinity with a variable (commonly \( b \)) and then computing its limit as \( b \to \infty \). This approach helps detail how the integral behaves as it stretches towards infinity. In our specific integral, replacing the upper limit with \( b \) transforms it into \( \int_{1}^{b} \frac{10}{x} \, dx \). Once integrated and simplified, you evaluate the limit \( \lim_{b \to \infty} 10 \ln |b| \), which diverges, meaning this particular integral's values keep increasing indefinitely.

Antiderivative

Finding the antiderivative is an essential step in solving integrals. It refers to reversing the process of differentiation. In simpler terms, if you differentiate a function and then integrate it, you should retrieve the original function. For the function \( \frac{10}{x} \), the antiderivative is \( 10 \ln |x| \). Calculating this is straightforward once you know the basic rules of logarithmic differentiation, where the derivative of \( \ln |x| \) is \( \frac{1}{x} \).

• Using the antiderivative, you can then estimate the area under the curve from a start point up to \( b \), and then deduce the broader trend as it approaches infinity.
• This method ties together with limit evaluation, where we see that \( 10 \ln |b| - 10 \ln 1 \) approaches infinity as \( b \to \infty \), thereby demonstrating the integral's divergence.

Mastery of finding antiderivatives can significantly ease solving both definite and improper integrals. They connect the dots between differentiation and integration, allowing a deeper insight into how calculus functions interact.