We will be focused on the volumes of solids obtained by revolving a region around an axis. Letβs use the running example of the region bounded by the curves .
Consider the below illustrated revolution of this region, and the cross-section drawn from a horizontal line segment. Choose the most appropriate description of this illustration.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Consider the below illustrated revolution of this region, and the cross-section drawn from a vertical line segment. Choose the most appropriate description of this illustration.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Consider the below illustrated revolution of this region, and the cross-section drawn from a horizontal line segment. Choose the most appropriate description of this illustration.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Consider the below illustrated revolution of this region, and the cross-section drawn from a vertical line segment. Choose the most appropriate description of this illustration.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Region is rotated around the -axis; the cross-sectional area is determined by the line segmentβs -value.
Generally when solving problems without the aid of technology, itβs useful to draw your region in two dimensions, choose whether to use a horizontal or vertical line segment, and draw its rotation to determine the cross-sectional shape.
When the shape is a disk, this is called the disk method and we use one of these formulas depending on whether the cross-sectional area depends on or .
When the shape is a washer, this is called the washer method and we use one of these formulas depending on whether the cross-sectional area depends on or .
When the shape is a cylindrical shell, this is called the shell method and we use one of these formulas depending on whether the cross-sectional area depends on or .
Draw a vertical line segment in one region and its rotation around the -axis. Draw a horizontal line segment in the other region and its rotation around the -axis.
Draw a vertical line segment in one region and its rotation around the -axis. Draw a horizontal line segment in the other region and its rotation around the -axis.