Problem 20 - Applied Calculus

Question

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{0}^{\infty} e^{-t} d t $$

Step-by-Step Solution

Answer

The integral converges to 1.

Step 1: Understanding the Improper Integral

The integral given is \( \int_{0}^{\infty} e^{-t} \, dt \). This is an improper integral, meaning it has infinite limits of integration. We need to assess if it converges to a finite value or diverges.

Step 2: Rewrite with a Limit

To deal with the infinity in the integral, introduce a limit: \( \lim_{b \to \infty} \int_{0}^{b} e^{-t} \, dt \). This means we first solve the integral with a finite upper limit \( b \), and then take the limit as \( b \) approaches infinity.

Step 3: Evaluate the Integral

Find the antiderivative of \( e^{-t} \), which is \( -e^{-t} \). Evaluate the definite integral from 0 to \( b \): \( \left[ -e^{-t} \right]_{0}^{b} = -e^{-b} + e^{0} = 1 - e^{-b} \).

Step 4: Take the Limit

Now, take the limit: \( \lim_{b \to \infty} (1 - e^{-b}) \). Since \( e^{-b} \rightarrow 0 \) as \( b \rightarrow \infty \), the limit becomes \( 1 - 0 = 1 \).

Step 5: Conclusion about Convergence

Since the limit exists and equals 1, the integral converges. Hence, \( \int_{0}^{\infty} e^{-t} \, dt = 1 \).

Key Concepts

Convergence of Integrals

When dealing with improper integrals, the first question to address is whether they converge to a finite value or diverge. Convergence of an integral essentially means that the area under the curve, from one endpoint to another, results in a finite value. In our case, we are analyzing an integral with limits extending to infinity, which initially makes determining convergence challenging.

To assess convergence, we use the process of introducing limits, transforming a potentially infinite problem into a finite one that is manageable and measurable. Once transformed, if the evaluation of the integral leads to a finite result as the interval expands indefinitely, we can conclude that the integral converges. If the result tends toward infinity, we declare it divergent.

Consider the integral:

• Assess whether you can express the integral with finite boundaries in some way.
• If a direct calculation does not lead to a finite result, explore methods such as using limits or comparison tests to find convergence.

Finding convergence is pivotal because it reveals whether a calculated solution is relevant and applicable based on the infinite properties of the original setup.

Infinite Limits

Handling integrals with infinite limits requires a careful approach, as these integrals may stretch over an infinite interval. The key to working with infinite limits is to replace the infinite boundary with a limit approach, allowing us to solve the integral in a more conventional manner.

In the exercise given, since the upper limit extends to infinity, we convert the problem using a limit:

• Introduce a variable that represents the approaching infinite limit, say \(b\).
• Perform the integration with this variable, treating it as a finite number, \( \int_{0}^{b} e^{-t} \, dt \).
• Finally, take the limit as the variable approaches infinity, \( \lim_{b \to \infty} \int_{0}^{b} e^{-t} \, dt \).

This method provides a pathway to evaluate the integral's behavior as it extends towards infinity, by focusing initially on how it behaves over finite expansions. Ultimately, this reveals whether the integral accounts for an infinite span smoothly converging into a defined result or not.

Antiderivative Evaluation

Evaluating the antiderivative is a central step in solving integrals, especially when they are improper. The antiderivative describes the original function from which the integrand derived, allowing us to evaluate definite integrals precisely.

For the integral \( \int e^{-t} \, dt \), we seek an expression whose derivative gives \( e^{-t} \). The antiderivative is \(-e^{-t}\). To find the complete integral:

• Calculate the antiderivative over the given interval, from the lower to the newly introduced finite upper limit \(b\), \( \left[ -e^{-t} \right]_{0}^{b} \).
• Evaluate this by substituting the bounds, obtaining a result involving \(-e^{-b}\) and \(-e^{0}\).
• Recognize that \(e^{0} = 1\), yielding \(1 - e^{-b}\) for this evaluation.

Upon determining the antiderivative form, complete the solution by computing the eventual limit for infinite spans. This step ensures a robust understanding of how the original function behaves and approximates broader areas in calculus.