Explanation Given: • f ( x ) = x sin x • 0 ≤ x ≤ π • x − a x i s Key Concepts Used: If a region R in the plane is revolved about a given line, the resulting solid is called a solid of revolution. The line about which the region is revolved is called the axis of revolution. Volume of a solid which is obtained by revolution is found by various methods like disk method, washer method and shell method. For this problem we have to use disk method to find the volume of solid obtained by revolving f ( x ) = x sin x for 0 ≤ x ≤ π about the x − axis. Method of disks to find the volume of solid which is obtained by revolution: Let f be a continuous function on a interval [ a , b ] with f ( x ) ≥ 0 for all x ∈ [ a , b ] . Let R denote the region between the curve y = f ( x ) , the x − axis and the lines x = a a n d x = b . When this region in the plane revolves around x − axis, then it forms a solid S , whose circular cross sections of radius is f ( x ) . The area of the cross section of S at x is the area of a circle with radius f ( x ) . That means the area is A ( x ) = π [ f ( x ) ] 2 . The volume of solid which is obtained by revolution is V = π ∫ f ( x ) 2 ⋅ d x a b (or) V = π ∫ y 2 ⋅ d x a b Here in this problem, f ( x ) = x sin x , [ a , b ] = [ 0 , π ] , and the axis is x − a x i s . Substitute these values in the given formula to find the required volume. Formula Used: The volume of solid which is obtained by revolution is V = π ∫ f ( x ) 2 ⋅ d x a b . Integration by parts Formula: ∫ u ( x ) ⋅ v ' ( x ) ⋅ d x = u ( x ) ⋅ v ( x ) − ∫ u ' ( x ) ⋅ v ( x ) ⋅ d x . Calculation: Description: V = π ∫ f ( x ) 2 ⋅ d x a b Step 1 : Here in the problem given that f ( x ) = x sin x , [ a , b ] = [ 0 , π ] . Substitute these values in the given formula. V = π ∫ f ( x ) 2 ⋅ d x a b = π ∫ ( x sin x ) 0 π 2 ⋅ d x V = π ∫ x 2 ⋅ sin x ⋅ d x 0 π Step 2 : We have to do the integration of ∫ x 2 ⋅ sin x ⋅ d x by using integration by parts formula...

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Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.1, Problem 65E

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Volume of solid obtained by revolving given function for the provided limit about the given axis

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Chapter 8.1, Problem 64E

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Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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Volume of solid obtained by revolving given function for the provided limit about the given axis

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