Problem 46 - Calculus and its applications

Question

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{-\infty}^{0} e^{2 x} d x $$

Step-by-Step Solution

Answer

The improper integral is convergent, with a value of \( \frac{1}{2} \).

Step 1: Set Up the Improper Integral

We need to evaluate the improper integral \( \int_{-\infty}^{0} e^{2x} \; dx \). Since the lower limit of the integral is \(-\infty\), we rewrite it as a limit: \( \lim_{a \to -\infty} \int_{a}^{0} e^{2x} \; dx \).

Step 2: Evaluate the Definite Integral

Find the antiderivative of \( e^{2x} \), which is \( \frac{1}{2}e^{2x} \). Evaluate the definite integral from \( a \) to 0: \( \left[ \frac{1}{2}e^{2x} \right]_{a}^{0} = \frac{1}{2}e^{0} - \frac{1}{2}e^{2a} = \frac{1}{2} - \frac{1}{2}e^{2a} \).

Step 3: Take the Limit as a Approaches \(-\infty\)

Compute the limit: \( \lim_{a \to -\infty} \left( \frac{1}{2} - \frac{1}{2}e^{2a} \right) \). As \( a \to -\infty \), \( e^{2a} \) approaches 0 because the exponent becomes very negative, making \( \frac{1}{2}e^{2a} \to 0 \). Hence, the limit is \( \frac{1}{2} \).

Step 4: Conclusion on Convergence

Since the limit exists and is a finite number, the improper integral converges. The value of the integral is \( \frac{1}{2} \).

Key Concepts

Convergent Series

A convergent series is an infinite series whose terms approach a specific value as more and more terms are added. In simpler terms, the sum of its parts gets closer and closer to a finite value.
This concept is central to understanding improper integrals, where you deal with functions extending towards infinity.For any improper integral like \( \int_{-\infty}^{0} e^{2x} \, dx \), you are effectively adding up an infinite number of small areas under the curve. If these areas sum to a specific finite number, we call the integral **convergent**. Otherwise, it is divergent and would not settle on any finite value.Evaluating whether an integral is convergent requires:

• Rewriting the integrals as limits of definite integrals.
• Examining the behavior of these expressions as the bounds go to infinity or negative infinity.

The concept of convergence assures us that even infinite processes can sometimes be reliably summed up, much like finding the value of our exercise integral which sums up to \( \frac{1}{2} \).

Antiderivatives

An antiderivative is a reversal of differentiation. It is a function, \( F(x) \), whose derivative gives back a function you started with, often denoted as \( f(x)\).
For our exercise, where \( f(x) = e^{2x} \), the antiderivative is essential.Finding the antiderivative is the first step to solving definite integrals, including improper ones. For \( e^{2x} \):

• The antiderivative is \( \frac{1}{2}e^{2x} \).

Having this antiderivative allows for easy computation of the definite integral between limits, like moving from \( a \) to \( 0 \) in our problem. The effective integration process returns a specific expression that, through limits and further evaluation, reveals whether we have convergence and what its value is.
In essence, antiderivatives allow us to leap back from rates of change to a complete picture of an area or accumulation of values.

Limits in Calculus

Limits are crucial for determining the behavior of functions as they approach specific values or infinity. In calculus, limits help to handle values that grow without bound, as in improper integrals.
When you deal with an integral that extends to infinity, you cannot compute it directly. Instead, you set it up as a limit. For instance, \( \int_{-\infty}^{0} e^{2x} \, dx \) is rewritten as:\[ \lim_{a \to -\infty} \int_{a}^{0} e^{2x} \, dx \]This expression is not just solved instantly but rather evaluated to see how it behaves as \( a \) approaches \(-\infty\). With limits, you focus on the approach rather than set endpoints.
This is vital because many functions have undefined behavior at infinity. Through limits, calculus offers a tool to manage these unpredictabilities, essentially ensuring we can assess whether functions converge to a specific value, such as determining the integral's value converges to \( \frac{1}{2} \). By using limits, we unlock a deeper understanding of functions over their entire domain, even extending to infinity.