Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{1} e^{4 x} d x\)

Step-by-Step Solution

Answer

The integral converges and its value is \( \frac{1}{4}e^{4} \).

Step 1: Define the Improper Integral

The given integral is an improper integral because it has an infinite lower limit. We will evaluate it by taking the limit as a parameter approaches infinity. Our problem is to evaluate \( \int_{-\infty}^{1} e^{4x} \, dx \).

Step 2: Set Up the Limit

To handle the infinite lower limit, we introduce a parameter \( t \) and consider the limit: \( \lim_{t \to -\infty} \int_{t}^{1} e^{4x} \, dx \). This replaces the infinite bound with a finite parameter, allowing us to evaluate the integral.

Step 3: Integrate the Function

Evaluate the indefinite integral of \( e^{4x} \). The integral of \( e^{kx} \) with respect to \( x \) is \( \frac{1}{k}e^{kx} \). So, the integral of \( e^{4x} \) is \( \frac{1}{4}e^{4x} + C \), where \( C \) is a constant of integration.

Step 4: Apply the Limits of Integration

With the antiderivative \( \frac{1}{4}e^{4x} \), we apply the limits from \( t \) to 1. Use the Fundamental Theorem of Calculus to evaluate it: \[ \int_{t}^{1} e^{4x} \, dx = \left[ \frac{1}{4}e^{4x} \right]_{t}^{1} = \frac{1}{4}e^{4 \cdot 1} - \frac{1}{4}e^{4t}. \]

Step 5: Evaluate the Limit

Next, we take the limit as \( t \to -\infty \) of the expression obtained: \[ \lim_{t \to -\infty} \left( \frac{1}{4}e^{4} - \frac{1}{4}e^{4t} \right). \] As \( t \to -\infty \), \( e^{4t} \to 0 \), since the exponential function approaches zero for large negative exponents. This simplifies the limit to \( \frac{1}{4}e^{4} - 0 = \frac{1}{4}e^{4} \).

Step 6: Conclusion

Since the limit results in a finite number, the integral \( \int_{-\infty}^{1} e^{4x} \, dx \) converges, and the evaluated value of the integral is \( \frac{1}{4}e^{4} \).

Key Concepts

Limit Evaluation

When dealing with improper integrals, particularly those with infinite limits, the concept of limit evaluation becomes crucial. Here, the integral \( \int_{-\infty}^{1} e^{4x} \, dx \) presents an infinite lower limit. To evaluate such integrals, we transform the problem into one that is simpler and more solvable.
**Step-by-Step Explanation:**

• Introduce a parameter, say \( t \), to replace the infinite bound. This changes the integral to something manageable: \( \lim_{t \to -\infty}\int_{t}^{1} e^{4x} \, dx \). Here, \( t \) approaches negative infinity, allowing us to handle the previously infinite boundary.
• Once the integral is in this form, we can focus on solving an integral with finite bounds, which is what standard calculus formulas and techniques are designed to address.
• Evaluate the definite integral and then apply a limit as \( t \) approaches the extreme value (in this case, negative infinity), ensuring that the result accounts for the originally improper nature of the integral.

Understanding this step is key to tackling similar problems involving improper integrals.

Exponential Function

Exponential functions, such as \( e^{4x} \), play a vital role in calculus, particularly in integrals. The function \( e^{kx} \) has properties, including rapid growth or decay, depending on the exponent's sign.
**Relevant Properties:**

• The integral of \( e^{kx} \) is \( \frac{1}{k} e^{kx} + C \), where \( C \) represents the constant of integration. This antiderivative formula is essential because it tells us how to reverse the differentiation process for exponential functions.
• These functions are continuous and possess derivatives that are also exponential, which makes them both predictable and mathematically tractable.
• The sign of \( k \) determines whether the function represents growth (if positive) or decay (if negative). In improper integrals like \( \int_{-\infty}^{1} e^{4x} \, dx \), a changing exponent like \( 4x \) means that even as \( x \) decreases towards \(-\infty\), the exponential term \( e^{4x} \) will decrease rapidly towards zero, enabling the convergence of integrals.

Recognizing these attributes helps in understanding the behavior of exponential functions within calculus problems.

Convergence of Integrals

The concept of convergence refers to the situation where an integral sums to a finite value, even if at first it may appear otherwise due to infinite limits or unbounded integrands. **Understanding Convergence:**

• In the exercise, after setting up the integral with finite bounds using a limit, we next look at the result, \( \lim_{t \to -\infty} ( \frac{1}{4}e^{4} - \frac{1}{4}e^{4t} ) \).
• The approach of \( e^{4t} \) towards zero as \( t \to -\infty \) ensures that the subtraction yields a finite result: \( \frac{1}{4}e^{4} \).
• If the limit results in a specific number, the integral is said to converge. In practical terms, this means you can compute it as part of solving mathematical models or real-world scenarios.

Grasping convergence is crucial because it allows you to determine whether or not an integral has a meaningful evaluation and answer.