I That is integrals of the type Z 1 1 Z 1 1 Z 1 1 A) 3 dx B) 3 dx C) 2 1 x 0 x −∞ 4 + x 1 I Note that the function f (x) = x3 has a discontinuity at x = 0 and the F.T.C. does not apply to B. I Note that the limits of integration for integrals A and C describe intervals that are infinite in length and the F.T.C. does not apply. Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Improper Integrals R b In this section, we will extend the concept of the definite integral a f (x)dx to functions with an infinite discontinuity and to infinite intervals. Annette Pilkington Improper Integrals 1 I Note that the function f (x) = x3 has a discontinuity at x = 0 and the F.T.C. does not apply to B. I Note that the limits of integration for integrals A and C describe intervals that are infinite in length and the F.T.C. does not apply. Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Improper Integrals R b In this section, we will extend the concept of the definite integral a f (x)dx to functions with an infinite discontinuity and to infinite intervals. I That is integrals of the type Z 1 1 Z 1 1 Z 1 1 A) 3 dx B) 3 dx C) 2 1 x 0 x −∞ 4 + x Annette Pilkington Improper Integrals I Note that the limits of integration for integrals A and C describe intervals that are infinite in length and the F.T.C. does not apply. Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Improper Integrals R b In this section, we will extend the concept of the definite integral a f (x)dx to functions with an infinite discontinuity and to infinite intervals. I That is integrals of the type Z 1 1 Z 1 1 Z 1 1 A) 3 dx B) 3 dx C) 2 1 x 0 x −∞ 4 + x 1 I Note that the function f (x) = x3 has a discontinuity at x = 0 and the F.T.C. does not apply to B. Annette Pilkington Improper Integrals Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Improper Integrals R b In this section, we will extend the concept of the definite integral a f (x)dx to functions with an infinite discontinuity and to infinite intervals. I That is integrals of the type Z 1 1 Z 1 1 Z 1 1 A) 3 dx B) 3 dx C) 2 1 x 0 x −∞ 4 + x 1 I Note that the function f (x) = x3 has a discontinuity at x = 0 and the F.T.C. does not apply to B. I Note that the limits of integration for integrals A and C describe intervals that are infinite in length and the F.T.C. does not apply. Annette Pilkington Improper Integrals Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Infinite Intervals An Improper Integral of Type 1 R t (a) If a f (x)dx exists for every number t ≥ a, then Z 1 Z t f (x)dx = lim f (x)dx a t!1 a provided that limit exists and is finite. R b (c) If t f (x)dx exists for every number t ≤ b, then Z b Z b f (x)dx = lim f (x)dx −∞ t→−∞ t provided that limit exists and is finite. R 1 R b The improper integrals a f (x)dx and −∞ f (x)dx are called Convergent if the corresponding limit exists and is finite and divergent if the limit does not exists. R 1 R a (c) If (for any value of a) both a f (x)dx and −∞ f (x)dx are convergent, then we define Z 1 Z a Z 1 f (x)dx = f (x)dx + f (x)dx −∞ −∞ a If f (x) ≥ 0, we can give theAnnette definite Pilkington integralImproper above Integrals an area interpretation. R 1 1 R t I By definition 1 x dx = limt!1 1 1=x dx ˛t I = limt!1 ln x˛ = limt!1(ln t − ln 1) ˛1 I = limt!1 ln t = 1 R 1 1 I The integral 1 x dx diverges. Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Infinite Intervals Z 1 Z t Z b Z b f (x)dx = lim f (x)dx f (x)dx = lim f (x)dx a t!1 a −∞ t→−∞ t If f (x) ≥ 0, we can give the definite integral above an area interpretation; namely that if the improper integral converges, the area under the curve on the infinite interval is finite. Example Determine whether the following integrals converge or diverge: Z 1 1 Z 1 1 dx; 3 dx; 1 x 1 x Annette Pilkington Improper Integrals ˛t I = limt!1 ln x˛ = limt!1(ln t − ln 1) ˛1 I = limt!1 ln t = 1 R 1 1 I The integral 1 x dx diverges. Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Infinite Intervals Z 1 Z t Z b Z b f (x)dx = lim f (x)dx f (x)dx = lim f (x)dx a t!1 a −∞ t→−∞ t If f (x) ≥ 0, we can give the definite integral above an area interpretation; namely that if the improper integral converges, the area under the curve on the infinite interval is finite. Example Determine whether the following integrals converge or diverge: Z 1 1 Z 1 1 dx; 3 dx; 1 x 1 x R 1 1 R t I By definition 1 x dx = limt!1 1 1=x dx Annette Pilkington Improper Integrals I = limt!1 ln t = 1 R 1 1 I The integral 1 x dx diverges. Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Infinite Intervals Z 1 Z t Z b Z b f (x)dx = lim f (x)dx f (x)dx = lim f (x)dx a t!1 a −∞ t→−∞ t If f (x) ≥ 0, we can give the definite integral above an area interpretation; namely that if the improper integral converges, the area under the curve on the infinite interval is finite. Example Determine whether the following integrals converge or diverge: Z 1 1 Z 1 1 dx; 3 dx; 1 x 1 x R 1 1 R t I By definition 1 x dx = limt!1 1 1=x dx ˛t I = limt!1 ln x˛ = limt!1(ln t − ln 1) ˛1 Annette Pilkington Improper Integrals R 1 1 I The integral 1 x dx diverges. Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Infinite Intervals Z 1 Z t Z b Z b f (x)dx = lim f (x)dx f (x)dx = lim f (x)dx a t!1 a −∞ t→−∞ t If f (x) ≥ 0, we can give the definite integral above an area interpretation; namely that if the improper integral converges, the area under the curve on the infinite interval is finite. Example Determine whether the following integrals converge or diverge: Z 1 1 Z 1 1 dx; 3 dx; 1 x 1 x R 1 1 R t I By definition 1 x dx = limt!1 1 1=x dx ˛t I = limt!1 ln x˛ = limt!1(ln t − ln 1) ˛1 I = limt!1 ln t = 1 Annette Pilkington Improper Integrals Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Infinite Intervals Z 1 Z t Z b Z b f (x)dx = lim f (x)dx f (x)dx = lim f (x)dx a t!1 a −∞ t→−∞ t If f (x) ≥ 0, we can give the definite integral above an area interpretation; namely that if the improper integral converges, the area under the curve on the infinite interval is finite. Example Determine whether the following integrals converge or diverge: Z 1 1 Z 1 1 dx; 3 dx; 1 x 1 x R 1 1 R t I By definition 1 x dx = limt!1 1 1=x dx ˛t I = limt!1 ln x˛ = limt!1(ln t − ln 1) ˛1 I = limt!1 ln t = 1 R 1 1 I The integral 1 x dx diverges. Annette Pilkington Improper Integrals R 1 1 R t 1 I By definition 1 x3 dx = limt!1 1 x3 dx −2 ˛t x ˛ I = limt!1 −2 ˛1 “ −1 1 ” I = limt!1 2t2 + 2 1 1 I = 0 + 2 = 2 . R 1 1 1 I The integral 1 x3 dx converges to 2 . Improper Integrals Infinite Intervals Area Interpretation Theorem 1 Functions with infinite discontinuities Comparison Test Comparison Test Infinite Intervals Z 1 Z t Z b Z b f (x)dx = lim f (x)dx f (x)dx = lim f (x)dx a t!1 a −∞ t→−∞ t Example Determine whether the following integrals converge or diverge: Z 1 1 Z 1 1 dx; 3 dx; 1 x 1 x Annette Pilkington Improper Integrals −2 ˛t x ˛ I = limt!1 −2 ˛1 “ −1 1 ” I = limt!1 2t2 + 2 1 1 I = 0 + 2 = 2 .