Explanation Given information: Theright circular cone is generated when the region bounded by the line y = x and the vertical lines x = 0 and x = r is revolved about the x-axis. 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. Proof: The right circular cone is generated when the region bounded by the line y = x and the vertical lines x = 0 and x = r is revolved about the x-axis. Area of the regionisgiven by A r e a = ∫ 0 r [ f ( x ) − g ( x ) ] d x = ∫ 0 r [ x − 0 ] d x = ( x 2 2 ) 0 r = r 2 2 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 = | y ¯ − ( 0 ) | is the vertical distance from the centroid to the axis of rotation, x-axis ( y = 0 ) . 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 ) ] d x a n d y ¯ = M x

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 6.5, Problem 55PS

To determine

To prove:The volume of the right circular cone is V=13πr3 using the theorem of Pappus.

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Chapter 6.5, Problem 54PS

Chapter 6.5, Problem 56PS

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Chapter 6 Solutions

Calculus

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To prove: The volume of the right circular cone is V = 1 3 π r 3 using the theorem of Pappus.

Solution Summary: The author proves the volume of the right circular cone using the theorem of Pappus.

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To prove:The volume of the right circular cone is V=13πr3 using the theorem of Pappus.

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Chapter 6 Solutions