Overview
This lecture introduces improper integrals, covering when standard integration rules fail, how to approach integrals with infinite intervals or unbounded functions, and how to determine convergence or divergence using limits and comparison.
Proper vs. Improper Integrals
- A definite integral ∫ₐᵇ f(x) dx requires f(x) bounded and finite on [a, b].
- Improper integrals occur if the interval is infinite or f(x) is unbounded/discontinuous within the interval.
- There are two cases:
- Case 1: Infinite interval (e.g., ∫₁^∞ f(x) dx).
- Case 2: f(x) is not bounded within [a, b] (e.g., vertical asymptotes).
Handling Infinite Intervals
- Replace infinity with a variable (e.g., B), evaluate the definite integral, and take the limit as B approaches infinity.
- For ∫₁^∞ (1/x²) dx, rewrite as lim₍ᴮ→∞₎ ∫₁ᴮ (1/x²) dx.
- Calculate the integral, substitute bounds, and evaluate the limit.
Handling Infinite Discontinuities
- If the problem is at a bound (e.g., at a or b), approach the problematic point with a limit from the appropriate side.
- For ∫₀⁹ (1/√x) dx, rewrite as lim₍ᶜ→0⁺₎ ∫𝑐⁹ (1/√x) dx and evaluate.
- If the discontinuity is at an interior point, split the interval and use separate limits for each section.
Convergence and Divergence
- An improper integral is convergent if the limit exists (finite area).
- It is divergent if the limit does not exist or is infinite.
- General result: ∫₁^∞ (1/xⁿ) dx converges if n > 1; diverges if n ≤ 1.
Comparison Test for Convergence
- If 0 ≤ f(x) ≤ g(x) on [a, ∞):
- If ∫ₐ^∞ g(x) dx converges, then ∫ₐ^∞ f(x) dx also converges.
- If ∫ₐ^∞ f(x) dx diverges, then ∫ₐ^∞ g(x) dx also diverges.
- Useful when an integral is too complex to solve directly.
Common Calculation Steps
- Identify if integral is improper (infinite interval or unbounded function).
- Rewrite with limits (use B for upper infinity, C for discontinuity).
- Compute antiderivative and substitute bounds.
- Evaluate the limit.
Key Terms & Definitions
- Improper Integral — Integral with infinite limits or unbounded integrand.
- Convergent — The improper integral results in a finite value.
- Divergent — The improper integral does not yield a finite value.
- Infinite Discontinuity — A point where f(x) goes to infinity within the interval.
- Comparison Test — Determines convergence by comparing to a known function.
Action Items / Next Steps
- Practice identifying improper integrals and rewriting them using limits.
- Review evaluation strategies for both types (infinite interval and infinite discontinuity).
- Complete assigned problems on improper integrals and convergence tests.