Improper Integrals Review

What are Improper Integrals?

• Improper integrals are definite integrals covering an unbounded area.
• Two types:
  1. Integrals with at least one endpoint extending to infinity.
  2. Integrals with finite endpoints but an unbounded function at one or more endpoints.
• Convergent vs Divergent:
  • Convergent: Integral has a finite value.
  • Divergent: Integral does not have a finite value.

Evaluating Improper Integrals

Type 1: Unbounded Endpoints

• Example: ( \int_{1}^{\infty} \frac{1}{x^2} , dx )
  • Expressed as a limit: ( \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^2} , dx )
  • Solution steps:
    1. Use the Fundamental Theorem of Calculus.
    2. Solve: ( \left[ -\frac{1}{x} \right]_{1}^{b} )
    3. Calculate limit: ( \lim_{b \to \infty} (1 - \frac{1}{b}) = 1 )_

Type 2: Unbounded Function

• Example: ( \int_{0}^{1} \frac{1}{x} , dx )
  • Expressed as a limit: ( \lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} , dx )
  • Solution steps:
    1. Find expression using the Fundamental Theorem of Calculus.
    2. Evaluate: ( \left[ \ln|x| \right]_{a}^{1} )
    3. Calculate limit: ( \lim_{a \to 0^+} (-\ln(a)) ); evaluate convergence._

Practice Problems

Problem 1: Unbounded Endpoint

• ( \int_{1}^{\infty} \frac{1}{x^3} , dx )
  • Possible answers:
    • ( \frac{1}{4} )
    • ( \frac{1}{2} )
    • ( 1 )
    • ( ) The integral diverges._

Problem 2: Unbounded Function

• ( \int_{0}^{8} \frac{1}{x^3} , dx )
  • Possible answers:
    • ( \frac{3}{2} )
    • ( 6 )
    • ( 8 )
    • The integral diverges._

Discussion and Questions

• Divergence and Convergence:
  • Divergent when output doesn't converge or approaches infinity.
  • Convergent when output approaches a finite value.
• Use of Limits:
  • Necessary for formal evaluations, especially in exams like AP Calculus.

Conclusion

• Understanding improper integrals involves recognizing unbounded areas/functions.
• Convergence and divergence characterize the integral's behavior.
• Practice with limits and the Fundamental Theorem of Calculus is crucial for solving improper integrals.