8.3 Improper Integrals

1. 8.3 Improper Integrals Integrals with infinite limits

2. Improper Integrals • If f(x) is continuous on [a,∞) then • If f(x) is continuous on (-∞,b] then • If f(x) is continuous on (-∞,∞) then Where c is any real number

3. Converge/diverge • If the limit is finite the integral converges and the limit is the value of the improper integral • If the limit fails to exist the integral diverges • For #3 the integral converges if BOTH improper integrals on the right-hand side converge

4. Direct Comparison Test • Let f and g be continuous on [a,∞) with 0≤f(x)≤g(x) for all x≥a

5. Limit Comparison Test If the positive functions f and g are continuous on [a,∞) and if then

6. Example 1: Determine whether the integral converges or diverges, if it converges evaluate the integral Separate the integral into 2 parts using a constant c Evaluate each integral one at a time

7. Evaluate the first integral making a substitution for -∞ Since the limit is a FINITE number, this integral CONVERGES

8. Now evaluate the second integral Since the limit is a FINITE number, this integral also CONVERGES

9. So…..

10. Example 2: Evaluate the integral or state that it diverges What is the function doing around 0 and /2? There is an infinite discontinuity at 0 Therefore the integral DIVERGES

11. Example 3: Use the direct comparison test to tell whether the integral diverges or converges Since this integral converges and 0≤f(x)≤2, then f(x) also converges

12. Example 4: Use the limit comparison test to determine if the integral converges or diverges Now use L’Hopitals Rule to find the limit Therefore the integral DIVERGES

13. Assignment: • Page 442 #1-45 every other odd