Problem 29 - Calculus Single Variable

Question

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{4} e^{x / 3} d x $$

Step-by-Step Solution

Answer

The integral converges, and its value is \( 3e^{4/3} \).

Step 1: Identify the integrand and limits

The given improper integral is \( \int_{-\infty}^{4} e^{x / 3} \, dx \). The integrand is \( e^{x/3} \), and the limits of integration are from \(-\infty\) to \(4\). Since the lower limit is \(-\infty\), we need to check the behavior of the integral as it approaches negative infinity.

Step 2: Rewrite the improper integral as a limit

Rewrite the improper integral as a limit by replacing the infinite limit with a variable \(t\): \[ \int_{- ext{infinity}}^{4} e^{x / 3} \, dx = \lim_{{t \to -\infty}} \int_{t}^{4} e^{x / 3} \, dx \]

Step 3: Compute the antiderivative

Find the antiderivative of \( e^{x/3} \). The antiderivative is \( abla e^{x/3} = 3e^{x/3} \).This is because the derivative of \( 3e^{x/3} \) with respect to \(x\) is \( e^{x/3} \).

Step 4: Evaluate the definite integral

Substitute the antiderivative back into the limit expression and evaluate the definite integral:\[ \int_{t}^{4} e^{x/3} \, dx = 3e^{x/3} \bigg|_{t}^{4} = 3e^{4/3} - 3e^{t/3} \]

Step 5: Evaluate the limit

Evaluate the limit as \( t \to -\infty \):\[ \lim_{{t \to -\infty}} \left( 3e^{4/3} - 3e^{t/3} \right) = 3e^{4/3} - 3 \times 0 = 3e^{4/3} \]The term \( 3e^{t/3} \) approaches zero because \( e^{t/3} \to 0 \) as \( t \to -\infty \).

Step 6: Determine convergence

The improper integral \( \int_{-\infty}^{4} e^{x / 3} \, dx \) converges because the limit \( 3e^{4/3} \) exists and is finite. Therefore, the integral converges.

Key Concepts

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function you started with. Finding an antiderivative is a central feature in calculus, especially when dealing with integration.

In the problem dealing with the integral \( \int e^{x/3} \, dx \), the antiderivative is found by identifying a function whose derivative would give \( e^{x/3} \). Here, it is found to be \( 3e^{x/3} \). Here's why:

• The derivative of \( e^{x/3} \) is \( \frac{1}{3}e^{x/3} \), given the chain rule of differentiation.
• Thus, multiplying \( e^{x/3} \) by 3 corrects for the \( \frac{1}{3} \), making \( 3e^{x/3} \) the antiderivative.

Understanding antiderivatives is essential because when you go backwards in calculus, from the derivative to the function itself, you're essentially finding antiderivatives.

Limits of Integration

In integration, the limits of integration are the values that define the start and end point of a region for which the integral is calculated. For definite integrals, you integrate between specific points along the x-axis.

In the case of improper integrals, one or both of the limits are often infinite or involve a discontinuity. Here, the limits of integration are from \(-\infty\) to 4, which means you need to calculate an integral that extends infinitely in one direction.

To approach solving this, we replace \(-\infty\) with a variable (in this case, \(t\)) and analyze how the integral behaves as \(t\) approaches \(-\infty\).

• Replace the infinite boundary: \( \int_{-\infty}^{4} e^{x/3} \, dx = \lim_{{t \to -\infty}} \int_{t}^{4} e^{x/3} \, dx \)
• This way, you transform the problem into something manageable by calculating a limit that represents an infinite process.

Handling the limits correctly ensures accurate results when determining convergence or divergence of the integral.

Convergence and Divergence

Convergence and divergence are fundamental concepts when discussing improper integrals. They help determine whether an integral has a finite value or not.

An improper integral converges if integrating over an infinite domain yields a finite result. Conversely, it diverges if it results in an infinite value or does not settle to any particular value.

• To establish convergence, we take the approach of computing a limit. For \( \int_{-\infty}^{4} e^{x/3} \, dx \), this involves expressing the improper integral in terms of limits.
• Set the expression: \( \lim_{{t \to -\infty}} \left( 3e^{4/3} - 3e^{t/3} \right) \).

In this exercise, as \( t \to -\infty \), \( e^{t/3} \to 0 \) because the exponential function decreases quickly towards zero. Therefore, the integral converges to a specific value, which is \( 3e^{4/3} \).

Convergence confirms that the area under the curve is finite within the specified limits.

Exponential Functions

Exponential functions are vital in mathematics, modeling growth and decay processes. They have the form \( f(x) = a^{x} \), where \( a \) is a constant. In calculus, exponential functions often appear in integration and differentiation problems.

The function \( e^{x/3} \) is a classic example, where \( e \) is Euler's number, approximately 2.718. This base makes the function's growth rate proportional to its current value, leading to unique properties:

• Rapid growth or decay, depending on positive or negative exponents.
• Self-similarity, which simplifies differentiation and integration.

When integrating exponential functions, understanding their behavior is critical:

• The antiderivative of \( e^{x/3} \) is \( 3e^{x/3} \), which follows from the exponential rule that the integral of \( e^{kx} \) is \( \frac{1}{k}e^{kx} \).
• In this exercise, the function decays as \( x \to -\infty \), demonstrating the exponential's diminishing effect over infinitely large negative inputs.

Recognizing these characteristics helps students predict how such functions will behave during integration.