Explanation Given information: The region bounded by the curve y = x , the x-axis, and the line x = 4 about the line x = − 1. Formula used: Theorem of Pappus: The solid generated by revolving a region R about a line outside its boundary has volume V = A s , where A is the area of R and s is the distance travelled by the centroid of R. Calculation: The region bounded by the curve y = x , the x-axis ( y = 0 ), and the line x = 4 . The limit of x between 0 to 4. Let f ( x ) = x and g ( x ) = 0 Area of the regionis given by A r e a = ∫ 0 4 [ f ( x ) − g ( x ) ] d x = ∫ 0 4 [ x − 0 ] d x = ( x 3 2 3 2 ) 0 4 = 2 3 ( 4 3 2 − 0 3 2 ) = 2 3 ( 8 ) = 16 3 By Pappu’s theorem, the volume of the solid generated by revolving the given region is given by V = A s = A ( 2 π d ) , w h e r e d = | x ¯ − ( − 1 ) | is the horizontal distance from the centroid to the axis of rotation, x = − 1 . The mass of R is given by m = ρ ∫ a b ( f ( x ) − g ( x ) ) d x And the centroid of R is the point ( x ¯ , y ¯ ) such that x ¯ = M y m = ∫ a b x [ f ( x ) − g ( x ) ] d x ∫ a b [ f ( x ) − g ( x

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ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.5, Problem 34PS

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To compute:The volume of the solids generated by revolving the given regions using the theorem of Pappus.

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Chapter 6.5, Problem 33PS

Chapter 6.5, Problem 35PS

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8) The volume of the region in the second quadrant bounded above by y=-x3, below by the x-axis and on the left by the line x=-1, revolved about the line x=-2 is: (3/5)pi C. units. ○ (2/5)pi C. units. ○ (5/3)pi C. units ○ (5/2)pi C. units.

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Answer the first 3 questions and if possible all 5

Chapter 6 Solutions

Calculus

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The volume of the solids generated by revolving the given regions using the theorem of Pappus.

Solution Summary: The author calculates the volume of the solid generated by revolving the given regions using the theorem of Pappus.

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Chapter 6 Solutions