Calculus I · Unit 3B · lesson
Cylindrical Shells
Learn Cylindrical Shells through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Area and volumeWhat this section is building
Learn Cylindrical Shells through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Area adds thin rectangles, slicing adds cross-sectional slabs, washers add annular slabs, and shells add thin cylindrical walls.
Sketch the region and axis, test vertical and horizontal slices, and choose the description that stays single-valued with the fewest interval splits.
Measuring a radius from the wrong curve, subtracting boundaries in the wrong order, or mixing a shell radius with its height.
Learning objectives
Use radius-times-height shell formulas and choose shells when they simplify the geometry.
Cylindrical Shells
Shells accumulate thin cylindrical walls
A strip parallel to the axis of rotation sweeps out a thin cylindrical shell. Its volume is approximately circumference times height times thickness: . The radius is the strip's distance from the axis, while the height is the length of the strip inside the region.
Shells are especially useful when washers would require solving for the other variable or splitting the region. They preserve the original orientation of many function descriptions. As with every volume method, draw one strip and imagine its rotation. If the strip is parallel to the axis, shells are natural; if it is perpendicular, disks or washers are natural. That geometric distinction is more reliable than memorizing which formula uses .
A thin cylindrical shell has approximate volume
Thus
Rotate under a parabola around the y-axis
Rotate the region under above the -axis for about the -axis. Shell radius is , height is :
u3b-shell-01Use shells to rotate the region under , , about the y-axis.
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Radius is , height is .
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Cylindrical shell from a vertical strip. A vertical strip at distance x from the y-axis forms a shell of radius x and height top-bottom. Show radius, height, circumference, and thickness. Do not confuse shell radius with function height.
Every relationship in cylindrical shell from a vertical strip uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Show radius, height, circumference, and thickness.
Cylindrical shell from a vertical strip. Show radius, height, circumference, and thickness.
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