1. 5.3 Copyright © 2014 Pearson Education, Inc. Improper Integrals OBJECTIVE Determine whether an improper integral is convergent or divergent. Solve applied problems involving improper integrals.

2. Slide 5- 2 Copyright © 2014 Pearson Education, Inc. DEFINITION: If the limit exists, then we say that the improper integral converges, or is convergent. If the limit does not exist, then we say that the improper integral diverges or is divergent. Thus, 5.3 Area Improper Integrals

3. Slide 5- 3 Copyright © 2014 Pearson Education, Inc. Example 1: Determine whether the following integral is convergent or divergent, and calculate its value if it is convergent: 5.3 Area Improper Integrals

4. Slide 5- 4 Copyright © 2014 Pearson Education, Inc. Example 1 (continued): 5.3 Area Improper Integrals

5. Slide 5- 5 Copyright © 2014 Pearson Education, Inc. Example 1 (concluded): As b approaches ∞, we know that e 2b approaches ∞. So, Thus, The integral is convergent. 5.3 Area Improper Integrals

6. Slide 5- 6 Copyright © 2014 Pearson Education, Inc. 5.3 Area Improper Integrals Quick Check 1 Determine whether the following integral is convergent or divergent, and calculate its value if it is convergent:

7. Slide 5- 7 Copyright © 2014 Pearson Education, Inc. 5.3 Area Improper Integrals Quick Check 1 Concluded We know that as Thus, The integral is convergent.

8. Slide 5- 8 Copyright © 2014 Pearson Education, Inc. DEFINITIONS: 5.3 Area Improper Integrals

9. Slide 5- 9 Copyright © 2014 Pearson Education, Inc. THEOREM 1 The accumulated present value of a continuous money flow into an investment at the rate of P dollars per year perpetually is given by where k is the rate and interest is compounded continuously. 5.3 Area Improper Integrals

10. Slide 5- 10 Copyright © 2014 Pearson Education, Inc. Example 2: Find the accumulated present value of an investment for which there is a perpetual continuous money flow of $2000 per year. Assume that the interest rate is 8%, compounded continuously. The accumulated present value is P/k or 5.3 Area Improper Integrals

11. Slide 5- 11 Copyright © 2014 Pearson Education, Inc. 5.3 Area Improper Integrals Quick Check 2 Find the accumulated present value of an investment for which there is a perpetual continuous money flow of $10,000 per year. Assume that the interest rate is 6%, compounded continuously. The accumulated present value of an investment is So,

12. Slide 5- 12 Copyright © 2014 Pearson Education, Inc. 5.3 Area Improper Integrals Section Summary An improper integral has infinity as one or both of its bounds and is evaluated using the limit: where is any real number.

13. Slide 5- 13 Copyright © 2014 Pearson Education, Inc. 5.3 Area Improper Integrals Section Summary Concluded The accumulated present value of a continuous money flow into an investment at the rate of dollars per year perpetually is given by where is the interest rate compounded continuously.