Explanation Solution: Statement: 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. Statement explanation: Consider a coordinate system in which the y-axis coincides with the axis of revolution. Consider the region R which is mentioned in the above picture where the area rectangle is Δ A and that it is located x units from the y-axis. The rectangle is revolving about the y-axis, it generates a shell of volume Δ V = 2 π x Δ A . Thus, by partitioning the region R into a number of rectangles and taking the limit of the sum of the volumes of all the related approximating shells as the norm of the partition approaches 0. That is given by V = lim ‖ P ‖ → 0 ∑ k = 1 n 2 π x k Δ A k = ∫ 2 π x d A = 2 π ∫ x d A

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

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 6.5, Problem 4PS

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To state and discuss:The volume theorem of Pappus.

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To state and discuss: The volume theorem of Pappus.

Solution Summary: The author explains the volume theorem of Pappus. The solid generated by revolving a region R has volume V=As and s is the distance travelled by the centroid of R

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Chapter 6 Solutions