Explanation Result used: Integration by parts: “Suppose u and v are differentiable functions, then ∫ a b u d v = u v | a b − ∫ a b v d u ”. Calculation: Consider the function f ( x ) = ln x . Since the region bounded by f ( x ) = ln x and the line y = 1 is revolved about x -axis, change the function in terms of independent variable y and compute the value of volume The revolution is about the x -axis; so, use the integration is in the shell method with respect to y . The shell runs parallel to the x -axis and radius of the shell is y where 0 ≤ y ≤ 1 . The shell extend from the y -axis to the curve y = ln x . Thus, the volume of the solid is V = ∫ 0 1 2 π y f ( y ) d y . y = ln x Take exponential for both sides, e y = x That is, the value of f ( y ) = e y . Substitute f ( y ) = e y in V = ∫ 0 1 2 π y f ( y ) d y and compute the volume of the solid. V = ∫ 0 1 2 π y e y d y = 2 π ∫ 0 1 y e y d y Let u = y and d v = e y d y

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 8.2, Problem 42E

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To find: The volume of the solid when the region bounded by f(x)=lnx and the line y=1 is revolved about the x-axis.

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Chapter 8.2, Problem 41E

Chapter 8.2, Problem 43E

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Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

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The volume of the solid when the region bounded by f ( x ) = ln x and the line y = 1 is revolved about the x -axis.

Solution Summary: The author calculates the volume of the solid when the region bounded by f(x)=mathrmlnx is revolved about the x -axis

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To find: The volume of the solid when the region bounded by f(x)=lnx and the line y=1 is revolved about the x-axis.

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Chapter 8 Solutions