Mathematics Foundations/10.4 Improper Integrals

Mathematics Foundations/Improper Integrals Overview Improper integrals extend definite integrals to cases with infinite limits or unbounded integrands. They are defined via limits. Convergence means the limit exists and is finite; otherwise, the integral diverges. Types of Improper Integrals 1. Unbounded interval: 2. Infinite discontinuity at c in [a,b]: is improper if is unbounded near . Define thumb|center|480px|Unbounded Interval: visualizing area accumulation as the upper limit tends to infinity; the curve’s tail behavior determines convergence. thumb|center|480px|Point Singularity: illustrating an integrand blowing up near c, with limits from left and right defining the integral. thumb|center|480px|Tail Behavior: depicting area far from the origin; emphasizes how decay rate controls convergence on [a,∞). thumb|center|480px|Comparison Test: f(x) bounded above by g(x); convergence inferred by comparing areas under two curves. thumb|center|480px|Limit Comparison: two functions with proportional tails; ratio approaching a finite nonzero limit. thumb|center|480px|p-Test (p>1): convergent example with steeply decaying tail for x^{-p} over [1,∞). thumb|center|480px|p-Test (p≤1): divergent case where the area under x^{-p} remains unbounded. thumb|center|480px|Splitting at Singularity: separating [a,c) and (c,b] and applying limit definitions to each side. Techniques 1. Find an antiderivative and take limits for the improper part. 2. Use comparison/limit comparison when direct evaluation is difficult. 3. Split integrals at singularities and treat each side separately. Worked Examples Unbounded interval: Evaluate . If , As , this converges iff i.e., . Infinite discontinuity: Evaluate . For , Taking , we get , hence convergent. Table: Summary of Improper Integral Types and Criteria + Improper Integral Summary Type Definition via limit Key convergence idea Unbounded interval Compare to a known tail (e.g., ) Negative infinity bound Use symmetry or comparison Point singularity Split at , test each side p-test converges iff Tail behavior determines convergence Exercises 1. Decide convergence of . 2. Evaluate for . 3. For which does converge? 4. Determine convergence of for . 5. Find such that converges.