Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{d x}{x^{0.99999}}\)

Step-by-Step Solution

Answer

The integral diverges because the expression tends to infinity with the limit.

Step 1: Identify the Integral Type

The integral given is \( \int_{1}^{\infty} \frac{d x}{x^{0.99999}} \), which is an improper integral because the upper limit of integration is infinity.

Step 2: Express the Integral in Terms of the Limit

Since the upper limit is infinity, express the integral using limits: \[ \lim_{b \to \infty} \int_{1}^{b} x^{-0.99999} \, dx \]

Step 3: Integrate the Function

Find the antiderivative of \( x^{-0.99999} \):\[ \int x^{-0.99999} \, dx = \frac{x^{1 - 0.99999}}{1 - 0.99999} = \frac{x^{0.00001}}{0.00001} + C \]

Step 4: Evaluate the Integral with Limits

Substitute the limits of integration:\[ \lim_{b \to \infty} \left( \frac{b^{0.00001}}{0.00001} - \frac{1^{0.00001}}{0.00001} \right) \]

Step 5: Assess Convergence

Evaluate the limit: \[ \lim_{b \to \infty} \frac{b^{0.00001}}{0.00001} \] The expression \( b^{0.00001} \) approaches infinity as \( b \to \infty \), making the entire integral diverge.

Key Concepts

Improper Integral Convergence

Improper integrals play a crucial role in calculus, especially when dealing with infinite limits. These integrals can sometimes be challenging to evaluate because of their infinite bounds. To determine if an improper integral converges, it's important to check its behavior as it approaches infinity.

• If the value of the integral approaches a finite number, the integral converges.
• If the value heads towards infinity, it diverges.

In the exercise provided, the upper limit of the integral is infinity, indicating that it is an improper integral. The first step is to express the integral as a limit, making it easier to analyze its behavior at infinity. Once expressed in terms of a limit, we integrate the function and then substitute the limits to assess convergence. The integral in question diverges because, as we evaluate the limit, the value approaches infinity.

Limit Evaluation in Calculus

Evaluating limits is a fundamental skill in calculus that helps us understand the behavior of functions as they approach specific points or infinity. When dealing with improper integrals, particularly ones with infinite limits, expressing the integral as a limit is essential before evaluation.

• The function under the integral sign is expressed as a limit: Ex: \(\lim_{b \to \infty} \int_{1}^{b} x^{-0.99999} \, dx\)
• After finding the antiderivative, substitute the bounds as the next step.

The main goal is to determine whether this limit yields a finite value. In this exercise, evaluating the limit showed that the integral did not converge, as the result tended towards infinity. Understanding this process aids in deciding whether an improper integral converges or diverges.

Antiderivative Calculation

Finding an antiderivative is crucial when solving definite and indefinite integrals. An antiderivative essentially reverses the differentiation process. To calculate the antiderivative of a function, you need to apply the standard rules of integration like power and exponential rules.

• For power functions, use the rule: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C, \text{ where } n eq -1\)

In the case of \( x^{-0.99999} \), the antiderivative is calculated using:\[\int x^{-0.99999} \, dx = \frac{x^{0.00001}}{0.00001} + C\]This expression is essential to compute definite integrals from given bounds. Understanding antiderivatives is key to handling a wide range of calculus problems related to integration.