Explanation Given information: Graph of y = ( x − 1 ) − 1 / 3 for x ≥ 2 . Formula used: ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x = L , where f ( x ) is a nonnegative function for x ≥ a and a be the fixed number. Proof: First, write the improper integral using given information and formula. By using the formula, The area defined by the improper integral ∫ 2 ∞ ( x − 1 ) − 1 / 3 d x . Let b > 2 , then lim b → ∞ [ ∫ 2 b ( x − 1 ) − 1 / 3 d x ] Now solve the improper integral by using the substitution method as follows: x − 1 = u ⇒ d x = d u When x = 2 , u = 1 ; when x = b , u = b − 1

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Chapter 9.6, Problem 20E

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To prove: The region under the graph of y=(x−1)−1/3 for x≥2 cannot be assigned any finite number as its area.

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Chapter 9.6, Problem 19E

Chapter 9.6, Problem 21E

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To prove: The region under the graph of y = ( x − 1 ) − 1 / 3 for x ≥ 2 cannot be assigned any finite number as its area.

Solution Summary: The author explains how the region under the graph of y=(x-1)-1/3 cannot be assigned any finite number as its area.

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To prove: The region under the graph of y=(x−1)−1/3 for x≥2 cannot be assigned any finite number as its area.

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Chapter 9 Solutions