Explanation Given: y = 12 x 2 + 5 x + 6 Formula used: [ F ( x ) ] a b = F ( b ) − F ( a ) Calculation: Consider the given function. y = 12 x 2 + 5 x + 6 Area of the given region is ∫ 0 4 12 x 2 + 5 x + 6 d x . Now, ∫ 12 x 2 + 5 x + 6 d x = 12 ∫ 1 x 2 + 5 x + 6 d x Since, x 2 + 5 x + 6 = ( x + 2 ) ( x + 3 ) Include one partial fraction for each factor and write 1 x 2 + 5 x + 6 = A x + 2 + B x + 3 Multiplying this equation with the least common denominator, we get 1 = A ( x + 3 ) + B ( x + 2 ) As equation is true for all x , we can substitute any convenient value for x to find A and B . To solve for A , let x = − 3 . A ( − 3 + 3 ) + B ( − 3 + 2 ) = 1 − B = 1 B = − 1 To solve for B , let x = − 2 . A ( − 2 + 3 ) + B ( − 2 + 2 ) = 1 A = 1 So, the decomposition is 1 x 2 + 5 x + 6 = 1 x + 2 + − 1 x + 3 = 1 x + 2 − 1 x + 3 Thus, ∫ 12 x 2 + 5 x + 6 d x = 12 ∫ ( 1 x + 2 − 1 x + 3 ) d x Use the basic integration rule: ∫ [ f ( u ) ± g ( u ) ] d u = ∫ f ( u ) d u ± ∫ g ( u ) d u Then, ∫ 12 x 2 + 5 x + 6 d x = 12 ∫ 1 x + 2 d x − 12 ∫ 1 x + 3 d x Use the log rule

7th Edition

ISBN: 9781337553032

Author: Larson, Ron, Edwards, Bruce H.

Publisher: Cengage Learning,

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Chapter 8.5, Problem 41E

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To calculate: The area of the given region y=12x2+5x+6

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Chapter 8.5, Problem 40E

Chapter 8.5, Problem 42E

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The area of the given region y = 12 x 2 + 5 x + 6

Solution Summary: The author calculates the area of the given region by multiplying the equation with the least common denominator.

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