Problem 41 - Calculus for Biology and Medicine

Question

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{0}^{\infty} e^{-x^{2} / 2} d x $$

Step-by-Step Solution

Answer

The integral converges by comparison with \( \int_{0}^{\infty} e^{-x} \, dx \).

Step 1: Identify the Integrand

The given integrand is \( e^{-x^2/2} \). We need to find a comparison function for it to determine the convergence of the integral.

Step 2: Choose a Comparison Function

We choose the function \( f(x) = e^{-x} \) as the comparison function because it is simpler and has a known behavior with respect to convergence on the interval \( [0, \infty) \).

Step 3: Compare the Functions

For \( x \geq 0 \), \( e^{-x^2/2} \leq e^{-x} \) because \( x^2/2 \geq x \) for \( x \geq 0 \). Thus, the integrand is dominated by \( e^{-x} \).

Step 4: Analyze Convergence of Comparison Integral

The integral \( \int_{0}^{\infty} e^{-x} \, dx \) is known to converge because it is an exponential function that rapidly approaches zero as \( x \to \infty \).

Step 5: Apply Comparison Test

Since \( e^{-x^2/2} \leq e^{-x} \) and \( \int_{0}^{\infty} e^{-x} \, dx \) converges, we can conclude by the comparison test that \( \int_{0}^{\infty} e^{-x^2/2} \, dx \) also converges.

Key Concepts

Comparison Test

The comparison test is a handy tool to determine the convergence of improper integrals. When faced with an integral that is difficult to directly evaluate, you can compare it to a simpler, known integral. Here’s how:

• Choose a Comparison Function: Pick a function that is simpler, often because it is easier to integrate or its convergence properties are known.
• Dominate the Original Function: Ensure your chosen function is greater than or equal to the original function over the interval in question. For the convergence assessment of the integral, it's important that the comparison function stays larger or equal because you'll use it to "dominate" the behavior of the original function to get a mathematical insight into its behavior.
• Apply the Test: If the comparison function's integral converges and it bounds the original function from above, then the original function's integral also converges.

In our current exercise, we used the function \( e^{-x} \) as our comparison benchmark because of its well-known convergence behavior over the interval \([0, \infty)\). Its integrability and rapid decay help us confirm that the original integral \( \int_{0}^{\infty} e^{-x^2/2} \, dx \) also converges.

Convergence of Improper Integrals

Improper integrals involve limits to infinity or undefined points. Understanding their convergence is crucial in calculus, and it forms a foundational aspect of the subject. These types of integrals can sometimes extend from negative to positive infinity or might have boundaries that cause the integrand to become undefined at some point.

• Identify the Limits: Check the limits of integration. If they are infinite or cause the function to become undefined, you have an improper integral.
• Consider Limit Behavior: You can determine convergence by checking how the function behaves as it approaches these troublesome points or as the ends of the interval expand.Compare to a Simpler Function: Use simpler functions to compare behavior. If the simpler function behaves and integrates well, so might your original function.

The provided exercise involves \( e^{-x^2/2} \) from 0 to infinity, making it improper due to the infinite upper limit. By comparing it to a compatible function and applying the comparison test, you confirm its convergence.

Exponential Functions

Exponential functions are a pivotal concept in calculus due to their distinctive growing and decaying characteristics. An exponential function can take the form \( e^x \), or variations involving negatives such as \( e^{-x} \).

• Decay and Growth: The function \( e^{-x} \) features exponential decay, which means it rapidly approaches zero as \( x \) increases. This property makes it exceptionally useful for comparison in integrals.
• Integration Properties: The exponential function, due to its simple derivative and integral (as both are proportional to \( e^x \) itself), simplifies many calculus operations.
• Use in Calculus: Exponential functions often appear in natural growth processes, decay processes, and in solving differential equations.

In our example, the function \( e^{-x^2/2} \) exhibits exponential decay on the interval \([0, \infty)\) when compared to \( e^{-x} \). This demonstrates why choosing a comparison function like \( e^{-x} \) is helpful; it not only reflects similar decay behavior but is analytically easier to handle.