A Twisted Solid A square of side length s lies in a plane...

A Twisted Solid A square of side length $s$ lies in a plane perpendicular to a line $L .$ One vertex of the square lies on $L .$ As this square moves a distance $h$ along $L,$ the square turns one revolution about $L$ to generate a corkscrew-like column with square cross sections. (a) Find the volume of the column. (b) Writing to Learn What will the volume be if the square turns twice instead of once? Give reasons for your answer.

A Twisted Solid A square of side length $s$ lies in a plane
perpendicular to a line $L .$ One vertex of the square lies on $L .$
As this square moves a distance $h$ along $L,$ the square turns one
revolution about $L$ to generate a corkscrew-like column with
square cross sections.
(a) Find the volume of the column.
(b) Writing to Learn What will the volume be if the square
turns twice instead of once? Give reasons for your answer.

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Key Concepts

Constant Cross Section

This concept refers to situations where the shape of the cross-sectional area remains unchanged as the solid is generated. In such cases, regardless of how the shape is oriented or rotated along its path, the total volume can be computed by multiplying the area of the cross section by the distance it travels.

Invariance of Volume Under Twisting

This idea highlights that if a cross-sectional shape rotates during the sweep, its volume remains unchanged because the rotation does not affect the overall area of the cross section or the distance covered. As a result, even a twisted solid, such as one generated by a rotating square, maintains the same volume as one without any rotation.

Method of Slicing (Integration)

This method involves dividing a solid into many very thin slices perpendicular to a chosen axis and then summing—or integrating—the volumes of these slices. When the cross section is constant, the integration simplifies to multiplying the cross-sectional area by the length of the solid, providing a straightforward approach to volume calculation.

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