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Understanding Improper Integrals: Convergence and Divergence Explained

Dive into the concept of improper integrals, where we encounter functions or limits that are infinite. This lesson, presented by Greg Kelly from Hanford High School, focuses on how to evaluate integrals of continuous functions over closed intervals, even when they exist at undefined points. Through illustrative examples, we explore left-hand and right-hand limits to determine whether integrals converge or diverge. Gain a clear understanding of how to handle functions with asymptotes and the impact of exponents on integral behavior.

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Understanding Improper Integrals: Convergence and Divergence Explained

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Presentation Transcript

  1. 8.4 day one Improper Integrals Greg Kelly, Hanford High School, Richland, Washington

  2. Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals. Until now we have been finding integrals of continuous functions over closed intervals.

  3. Example 1: The function is undefined at x = 1 . Can we find the area under an infinitely high curve? Since x = 1 is an asymptote, the function has no maximum. We could define this integral as: (left hand limit) We must approach the limit from inside the interval.

  4. Rationalize the numerator.

  5. This integral converges because it approaches a solution.

  6. Example 2: (right hand limit) We approach the limit from inside the interval. This integral diverges.

  7. The function approaches when . Example 3:

  8. If then gets bigger and bigger as , therefore the integral diverges. If then b has a negative exponent and , therefore the integral converges. Example 4: (P is a constant.) What happens here? p

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