Explanation Given information: In 2012 , the population density of a city t miles from the city centre was 120 e − 0.65 t thousand people per square mile. Calculation: D ( t ) be the density of population (in persons per mile) at distance t miles from the city center. Therefore, D ( t ) = 120 e − 0.65 t Ring includes all people who live between a and b miles from the city center. Hence, a = 3 and b = 5 Thus, the total population in a ring around the city center = ∫ a b 2 π t D ( t ) d t The number of people (in thousands) who lived between 3 and 5 miles from the city center is ∫ 0 5 2 π t ( 120 e − 0.65 t ) d t = 240 π ∫ 0 5 t e − 0.65 t d t . f ( t ) = t , f ′ ( t ) = 1 , G ( t ) = ∫ e − 0.65 t d t = − e − 0.65 t 0.65 By using the integration by parts rule, 240 π ∫ 3 5 t ( e − 0.65 t ) d t = 240 π [ ( t ) ( − e − 0.65 t 0.65 ) | 3 5 − ∫ 3 5 ( 1 ) ( − e − 0

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Chapter 9.5, Problem 10E

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The number of people (in thousands) who lived between 3 and 5 miles from the city center, where in 2012, the population density of a city t miles from the city center was 120 e−0.65t thousand people per square mile.

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Chapter 9.5, Problem 9E

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The number of people (in thousands) who lived between 3 and 5 miles from the city center, where in 2012 , the population density of a city t miles from the city center was 120 e − 0.65 t thousand people per square mile.

Solution Summary: The author calculates the total population in a ring around the city center by using the integration by parts rule.

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The number of people (in thousands) who lived between 3 and 5 miles from the city center, where in 2012, the population density of a city t miles from the city center was 120 e−0.65t thousand people per square mile.

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