SECTION 8.4 IMPROPER INTEGRALS

SECTION 8.4 IMPROPER INTEGRALS Learning Targets: –I can evaluate Infinite Limits of Integration –I can evaluate the Integral –I can evaluate integrands with Infinite Discontinuities –I can use tests for Convergence and Divergence

INFINITE LIMITS OF INTEGRATION Definition: Improper Integrals with Infinite Integration Limits Integrals with infinite limits of integration are improper integrals. 1.If f(x) is continuous on [a, ), then 2.If f(x) is continuous on (, b], then 3.If f(x) is continuous on (, ), then where c is any real number.

DEFINITION Convergent – In parts 1 and 2, an improper integral converges if the limit is finite. The value of the limit is the value of the improper integral. In part 3, if both parts on the right-hand side converge, then the improper integral on the left converges. Divergent – In parts 1 and 2, if the limit of the fails to exist, the improper integral diverges. In part 3, if one or both of the right-hand integrals diverge, then the entire improper integral is divergent.

EXAMPLE 1 Express the improper integral in terms of limits of definite integrals and then evaluate the integral. Split at 0 like in part 3. Because the 2 nd part diverges, the entire integral diverges.

EXAMPLE 2 Does the improper integral converge or diverge? Use part 1 (infinity on top) The integral diverges.

EXAMPLE 3 Evaluate or state that it diverges. Partial Fractions YEAH! Solve for A and B… Converges!

EXAMPLE 4 Evaluate or state that it diverges IBP or tabular integration (you choose!) xe -x 1-e -x 0e -x Therefore, the integral converges

EXAMPLE 5 Evaluate HOORAY for flash cards!! Therefore, the integral converges

INTEGRANDS WITH INIFINITE DISCONTINUITIES Another type of improper integral happens when our integrand has an essential discontinuity (vertical asymptote) at a limit of integration or some point between the limits of integration. Definition: Improper Integrals with Infinite Discontinuities Integrals of functions that become infinite at a point within the interval of integration are improper integrals. 1.If f(x) is continuous on (a, b], then 2.If f(x) is continuous on [a, b), then 3.If f(x) is continuous on [a, c) and (c, b] then where c is any real number. Note: The same rules from above apply for convergence and divergence of functions with infinite discontinuities.

EXPLORATION: INVESTIGATING 1. Explain why these integrals are improper if p > 0. 2.Show that the integral diverges if p = Show that the integral diverges if p > Show that the integral converges if 0 < p < 1. Infinite discontinuity at x = 0 Because (-p+1) < 0 Therefore, convergent!

EXAMPLE 6 Evaluate Vertical asymptote at x = 1 Using part 3, split the integral at x = 1 This confirms the p-series test with a p < 1!

EXAMPLE 7 Evaluate Vertical asymptote at x = 2 Using part 3, split the integral at x = 2 Integral diverges This confirms the p-series test with a p = 1!

8.4 HOMEWORK # 6, 9, 12, 15, 18, 24, 26, 38,