AP Calculus BC 8.4 Improper Integrals Objective: able to use limits to evaluate improper integrals; to determine the convergence or divergence of improper integrals
1. Evaluate the following definite integral .
2. Evaluate the following definite integral , R is a real number greater than zero .
3. Evaluate the following improper integral .
Improper Integrals with Infinite Integration Limits
In parts 1 and 2, if the limit is finite the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of the equation converges if both improper integrals on the right-hand side converge, otherwise it diverges and has no value.
4. Evaluate . 5. Evaluate
6. Why is the following integral improper?
Improper Integrals with Infinite Discontinuities
In parts 1 and 2, if the limit is finite the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist the improper integral diverges. In part 3, the integral on the left-hand side of the equation converges if both integrals on the right-hand side have values, otherwise it diverges.
7. Evaluate the improper integral or state that it diverges .
Comparison Test
8. Use the comparison test to determine the convergence or divergence of .
Hint: HW Problem #37, P. 468 Recall: Lesson 3.8 & u-substitution. Let = √ , then let 1 + = 1 + √
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