Problem 67 - Applied Calculus

Question

Does \(\int_{0}^{\infty} 5 d x\) converge or diverge?

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Determine the Type of Integral

The given integral is an improper integral of the form \( \int_{0}^{\infty} 5 \, dx \). This is because the upper limit of integration is infinity.

Step 2: Set Up the Limit for the Improper Integral

We consider the integral as a limit: \[\lim_{b \to \infty} \int_{0}^{b} 5 \, dx\]This allows us to evaluate the improper integral as the limit of a proper integral.

Step 3: Calculate the Definite Integral

Compute the integral \( \int_{0}^{b} 5 \, dx \). \[\int_{0}^{b} 5 \, dx = \left[ 5x \right]_{0}^{b} = 5b - 5 \times 0 = 5b\] Thus, the result of the integral is \( 5b \).

Step 4: Evaluate the Limit

Take the limit as \( b \to \infty \) of the expression obtained in Step 3:\[\lim_{b \to \infty} 5b = \infty\]Since the result is infinity, the integral does not have a finite value.

Step 5: Conclusion

Since the result of evaluating the integral's limit is infinity, the improper integral \( \int_{0}^{\infty} 5 \, dx \) diverges.

Key Concepts

Convergence and Divergence of Integrals

In the realm of calculus, improper integrals are sought to determine whether they converge or diverge. Convergence means that the integral approaches a finite value, while divergence means it does not. An integral like \( \int_{0}^{\infty} 5 \, dx \) has an infinite upper limit, making it improper.

To assess convergence or divergence:

• Transform the improper integral into a limit. For example, \( \lim_{b \to \infty} \int_{0}^{b} 5 \, dx \).
• Evaluate this limit. If it approaches a finite number, the integral converges.
• If it tends towards infinity or fails to settle near any number, it diverges.

In this case, because \( 5b \) grows infinitely as \( b \to \infty \), our integral diverges. Understanding these principles is key to tackling problems involving infinite limits.

Evaluating Limits in Calculus

Calculus often involves calculating limits, which specify the value a function approaches as a variable approaches some point. When dealing with improper integrals, evaluating limits becomes indispensable.

For example, consider the expression \( \lim_{b \to \infty} 5b \). This shows how \( 5b \) behaves as \( b \) increases infinitely:

• If substitution were possible, direct evaluation would follow. However, limits often require deeper analysis.
• Using properties of limits, one checks if the expression stabilizes or decreases indefinitely.
• In the case of \( 5b \), as \( b \to \infty \), \( 5b \) clearly increases without bound, confirming the original integral diverges.

Grasping limit evaluation nurtures problem-solving skills, especially when traditional algebraic manipulation falls short.

Definite and Indefinite Integrals

Within calculus, integrals reign as vital tools. They are categorized into definite and indefinite integrals, each serving unique purposes.

• **Definite Integrals**: Evaluate to find the net area under a curve over a specific interval, providing a numerical result. For example, \( \int_{0}^{b} 5 \, dx = 5b \).
• **Indefinite Integrals**: Represent the antiderivative of a function, indicating a family of functions without fixed limits, such as \( \int 5 \, dx = 5x + C \) where \( C \) is a constant.

The step of transitioning an improper integral into a definite integral (with limits) is crucial, as it simplifies evaluating the integral and determining convergence. Starting from an indefinite perspective, one can specially tailor it into the limit form to further inspect its behavior.