Question

What are the two general ways in which an improper integral may occur?

Short answer

Question: Identify two general ways in which an improper integral may occur and provide an example for each. Answer: An improper integral may occur due to (1) an infinite interval of integration, such as integrating a function from negative infinity to positive infinity; example: ∫[-∞, ∞] e^(-x^2) dx, and (2) an integrand that approaches infinity within the interval of integration, which typically happens when the function has a vertical asymptote; example: ∫[0, 1] (1 / sqrt(x)) dx.

Step-by-step solution

1. Infinite interval of integration

An improper integral occurs when one or both limits of integration are infinite, such as integrating a function from negative infinity to positive infinity or from a finite value to infinity. Example: Consider the integral: ∫[-∞, ∞] e^(-x^2) dx Here, the limits of integration are infinite (-∞ and ∞), which makes this an improper integral.

2. An integrand that approaches infinity

An improper integral may also occur when the function being integrated has one or more points where the integrand approaches infinity within the interval of integration. This can happen when the function has a vertical asymptote within the interval. Example: Consider the integral: ∫[0, 1] (1 / sqrt(x)) dx Here, the integrand approaches infinity when x = 0, which makes this an improper integral.