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Calculus I · Unit 3B · lesson

Cylindrical Shells

Concept

Learning objectives

Use radius-times-height shell formulas and choose shells when they simplify the geometry.

Cylindrical Shells

Explanation

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: 2π(radius)(height)(thickness)2\pi(\text{radius})(\text{height})(\text{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 dxdx.

A thin cylindrical shell has approximate volume

2π(radius)(height)(thickness).2\pi(\text{radius})(\text{height})(\text{thickness}).

Thus

V=2π∫abr(x)h(x) dx.V=2\pi\int_a^b r(x)h(x)\,dx.
Worked example

Rotate under a parabola around the y-axis

Rotate the region under y=4−x2y=4-x^2 above the xx-axis for 0≤x≤20\le x\le2 about the yy-axis. Shell radius is xx, height is 4−x24-x^2:

V=2π∫02x(4−x2)dx=8π.V=2\pi\int_0^2x(4-x^2)dx=8\pi.
Interactive checku3b-shell-01

Use shells to rotate the region under y=4−x2y=4-x^2, 0≤x≤20\le x\le2, about the y-axis.

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Show hint

Radius is xx, height is 4−x24-x^2.

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Cylindrical shell from a vertical strip. Show radius, height, circumference, and thickness.
Read this graph as text

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.

Visual study

Cylindrical shell from a vertical strip. Show radius, height, circumference, and thickness.

After the explanation

Use the section idea

Reading lens

Let the axis and slice orientation determine every distance; top-minus-bottom, right-minus-left, radii, and shell height must all come from the same picture.

Mental model

Area adds thin rectangles, slicing adds cross-sectional slabs, washers add annular slabs, and shells add thin cylindrical walls.

Decision

Sketch the region and axis, test vertical and horizontal slices, and choose the description that stays single-valued with the fewest interval splits.

Common trap

Measuring a radius from the wrong curve, subtracting boundaries in the wrong order, or mixing a shell radius with its height.

Check yourself

Do the slice dimensions remain nonnegative on the full interval, and do their units multiply to area or volume?

Source & rights

Original instruction with traceable references.

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