Explanation Given information : ∫ 1 ∞ d x x 3 Formula Used: Negative exponent rule: 1 a m = a − m Power rule of integral: ∫ a b x n ⋅ d x = [ x n + 1 n + 1 ] a b Concept Involved: If ∫ a t f ( x ) ⋅ d x exists for every t > a , then ∫ a ∞ f ( x ) ⋅ d x = lim t → ∞ ∫ a t f ( x ) ⋅ d x provided the limit exists and is finite. If ∫ t b f ( x ) ⋅ d x exists for every t < b , then ∫ − ∞ b f ( x ) ⋅ d x = lim t → − ∞ ∫ t b f ( x ) ⋅ d x provided the limit exists and is finite. If a > 0 then ∫ a ∞ 1 x p d x is convergent if p > 1 and divergent if p ≤ 1 Calculation: Description Steps Convert the integral to a limit ∫ 1 ∞ d x x 3 = lim t → ∞ ∫ 1 t 1 x 3 ⋅ d x Rewriting the integral using the negative exponent rule lim t → ∞ ∫ 1 t x − 3 &

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

Author: STRAUSS MONTY J, TODA MAGDALENA DANIELE, SMITH KARL J

Publisher: Kendall Hunt Publishing

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Chapter 7.7, Problem 3PS

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3. We'd like to know the first time when the population reaches 7000 people. First, graph the function from part (a) on your calculator or Desmos. In the same window, graph the line y = 7000. Notice that you will need to adjust your window so that you can see values as big as 7000! Investigate the intersection of the two graphs. (This video shows you how to find the intersection on your calculator, or in Desmos just hover the cursor over the point.) At what value t> 0 does the line intersect with your exponential function? Round your answer to two decimal places. (You don't need to show work for this part.) (2 points)

Chapter 7 Solutions

Student's Solution and Survival Manual for Calculus

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Tofind : the value of improper integral if it converges or show that it diverges.

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Chapter 7 Solutions