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Section 8.8 Calculus BC AP/Dual, Revised ©2019 viet.dang@humbleisd.net
Improper Integrals Section 8.8 Calculus BC AP/Dual, Revised ©2019 1/27/2020 7:40 AM §8.8: Improper Integrals
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Improper Integration To define an improper integral, they must have either or both of these situations: They have an infinite interval of integration They have a discontinuity of the interval of integration 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 1 Determine whether 𝟓 𝟏𝟎 𝟏 𝒙−𝟐 𝟐 𝒅𝒙 is an improper integral. Does it have any discontinuities? If so, what are they? Is the discontinuity in the range from 𝒂,𝒃 ? 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 2 Determine whether 𝟐 𝟏𝟎 𝟏 𝒙−𝟐 𝟐 𝒅𝒙 is an improper integral. Does it have any discontinuities? If so, what are they? Is the discontinuity in the range from 𝒂,𝒃 ? 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 3 Determine whether −∞ ∞ 𝟏 𝒙−𝟐 𝟐 𝒅𝒙 is an improper integral. Does it have any discontinuities? If so, what are they? Is the discontinuity in the range from 𝒂,𝒃 ? 1/27/2020 7:40 AM §8.8: Improper Integrals
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Review Evaluate 𝟏 𝒆 𝟏 𝒙 𝒅𝒙 on the calculator. 1/27/2020 7:40 AM
§8.8: Improper Integrals
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Steps Make sure the integral is continuous functions over closed intervals Substitute 𝒂 and/or 𝒃 with the infinity (make it proper integral) and integrate Convergent: Approaching to a number If 𝒇 is continuous on the interval 𝒂,∞ , then 𝒂 ∞ 𝒇 𝒙 𝒅𝒙= 𝐥𝐢𝐦 𝒃→∞ 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 If 𝒇 is continuous on the interval −∞,𝒃 , then −∞ 𝒃 𝒇 𝒙 𝒅𝒙= 𝐥𝐢𝐦 𝒂→−∞ 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 Divergent: Approaching to ±∞ If 𝒇 is continuous on the interval (−∞,∞), then −∞ ∞ 𝒇 𝒙 𝒅𝒙= −∞ 𝒄 𝒇 𝒙 𝒅𝒙 + 𝒄 ∞ 𝒇 𝒙 𝒅𝒙 where 𝒄 is any real number 1/27/2020 7:40 AM §8.8: Improper Integrals
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Convergent Vs. Divergent
Convergent: Divergent: 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 4 Evaluate 𝟏 ∞ 𝟏 𝒙 𝒅𝒙 and determine whether the function is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 4 Evaluate 𝟏 ∞ 𝟏 𝒙 𝒅𝒙 and determine whether the function is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 4 Evaluate 𝟏 ∞ 𝟏 𝒙 𝒅𝒙 and determine whether the function is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 4 Evaluate 𝟏 ∞ 𝟏 𝒙 𝒅𝒙 and determine whether the function is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Next Three Examples… …Discover if you notice a noticeable pattern.
§8.8: Improper Integrals
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Example 5 Evaluate 𝟏 ∞ 𝟏 𝒙 𝟑/𝟐 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 5 Evaluate 𝟏 ∞ 𝟏 𝒙 𝟑/𝟐 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 6 Evaluate 𝟏 ∞ 𝟏 𝒙 𝟐 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 6 Evaluate 𝟏 ∞ 𝟏 𝒙 𝟐 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 7 Evaluate 𝟏 ∞ 𝟏 𝒙 𝟑 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Compare Examples 5 to 7 Example 2: 𝟏 ∞ 𝟏 𝒙 𝟑/𝟐 𝒅𝒙 Example 3: 𝟏 ∞ 𝟏 𝒙 𝟐 𝒅𝒙 Example 4: 𝟏 ∞ 𝟏 𝒙 𝟑 𝒅𝒙 1/27/2020 7:40 AM §8.8: Improper Integrals
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Rule of P-Series Improper Integrals
If 𝒂>𝟎 , then 𝒂 ∞ 𝟏 𝒙 𝑷 𝒅𝒙 is convergent if |𝒑|>𝟏 If 𝒂>𝟎 , then 𝒂 ∞ 𝟏 𝒙 𝑷 𝒅𝒙 is divergent if |𝒑|≤𝟏 If 𝒂=𝟏 and |𝒑|>𝟏, then 𝟏 ∞ 𝟏 𝒙 𝑷 𝒅𝒙 is converges to 𝟏 𝑷−𝟏 1/27/2020 7:40 AM §8.8: Improper Integrals
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Your Turn Evaluate 𝟏 ∞ 𝟓 𝒙 𝟑 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Your Turn Evaluate 𝟏 ∞ 𝟓 𝒙 𝟑 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 8 Evaluate 𝟎 𝟏 𝟏 𝟑 𝒙 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 9 Evaluate −∞ 𝟎 𝟏 𝟑−𝒙 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 9 Evaluate −∞ 𝟎 𝟏 𝟑−𝒙 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 9 Evaluate −∞ 𝟎 𝟏 𝟑−𝒙 𝒅𝒙 and determine whether the function is convergent or divergent. 1/27/2020 7:40 AM §8.8: Improper Integrals
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Natural Base Graph 1/27/2020 7:40 AM §8.8: Improper Integrals
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Several Hints 𝒏 ∞ =𝟎; Small/Big = 0 ∞ 𝒏 =𝟎; Big/Small = ±∞
𝐥𝐢𝐦 𝒙→−∞ 𝒆 𝒙 = 𝒆 −∞ =𝟎 𝐥𝐢𝐦 𝒙→∞ 𝒆 −𝒙 = 𝒆 −∞ =𝟎 ∞ 𝒏 =𝟎; Big/Small = ±∞ 𝐥𝐢𝐦 𝒙→∞ 𝒆 𝒙 = 𝒆 ∞ =∞ 𝐥𝐢𝐦 𝒙→−∞ 𝒆 −𝒙 = 𝒆 ∞ =∞ 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 10 Evaluate −∞ 𝟎 𝒆 𝒙/𝟒 𝒅𝒙 and determine if the improper integral is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 10 Evaluate −∞ 𝟎 𝒆 𝒙/𝟒 𝒅𝒙 and determine if the improper integral is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Your Turn Evaluate 𝟎 ∞ 𝒆 −𝒙 𝒅𝒙 and determine if the improper integral is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 11 Evaluate −∞ ∞ 𝒆 𝒙 𝟏+ 𝒆 𝟐𝒙 𝒅𝒙 and determine if the improper integral is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 11 Evaluate −∞ ∞ 𝒆 𝒙 𝟏+ 𝒆 𝟐𝒙 𝒅𝒙 and determine if the improper integral is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 11 Evaluate −∞ ∞ 𝒆 𝒙 𝟏+ 𝒆 𝟐𝒙 𝒅𝒙 and determine if the improper integral is convergent or divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 12 Let 𝒇 𝒙 = 𝒆 −𝟐𝒙 for 𝟎≤𝒙<∞, and let 𝑹 be the unbounded region in the first quadrant below the graph of 𝒇. Find the volume of the solid generated when 𝑹 is revolved around the 𝒙-axis 1/27/2020 7:40 AM §8.8: Improper Integrals
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Example 13 Solve −𝟑 𝟑 𝟏 (𝒙+𝟐) 𝟑 𝒅𝒙 1/27/2020 7:40 AM
§8.8: Improper Integrals
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Example 13 Solve −𝟑 𝟑 𝟏 (𝒙+𝟐) 𝟑 𝒅𝒙 1/27/2020 7:40 AM
§8.8: Improper Integrals
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AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝟏 ∞ 𝒙 𝒆 −𝒙 𝟐 𝒅𝒙 (A) − 𝟏 𝒆 (B) 𝟏 𝟐𝒆 (C) 𝟐 𝒆 (D) Answer is Divergent 1/27/2020 7:40 AM §8.8: Improper Integrals
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AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝟏 ∞ 𝒙 𝒆 −𝒙 𝟐 𝒅𝒙 Vocabulary Connections and Process Answer 1/27/2020 7:40 AM §8.8: Improper Integrals
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Assignment Worksheet 1/27/2020 7:40 AM §8.8: Improper Integrals
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