A Shell

1. A hard external covering of an animal.

• The calcareous or chitinous external covering of mollusks, crustaceans, and some other invertebrates.
• (by extension) Any mollusk having such a covering.
• (entomology) The exoskeleton or wing covers of certain insects.
• The conjoined scutes that constitute the "shell" (carapace) of a tortoise or turtle.
• The overlapping hard plates comprising the armor covering the armadillo's body.

2. The hard calcareous covering of a bird egg.
3. One of the outer layers of skin of an onion.
4. The restaurant served caramelized onion shells.

• (botany) The hard external covering of various plant seed forms.
• A pod containing the seeds of certain plants, such as the legume Phaseolus vulgaris.
• (in the plural) Husks of cacao seeds, a decoction of which is sometimes used as a substitute or adulterant for cocoa and its products such as chocolate.

5. (geology) The accreted mineral formed around a hollow geode.
6. (weaponry) The casing of a self-contained single-unit artillery projectile.
7. (weaponry) A hollow usually spherical or cylindrical projectile fired from a siege mortar or a smoothbore cannon. It contains an explosive substance designed to be ignited by a fuse or by percussion at the target site so that it will burst and scattered at high velocity its contents and fragments. Formerly called a bomb.
8. (weaponry) The cartridge of a breechloading firearm; a load; a bullet; a round.
9. (architecture) Any slight hollow structure; a framework, or exterior structure, regarded as not complete or filled in, as the shell of a house.
10. A garment, usually worn by women, such as a shirt, blouse, or top, with short sleeves or no sleeves, that often fastens in the rear.
11. A coarse or flimsy coffin; a thin interior coffin enclosed within a more substantial one.
12. (music) A string instrument, as a lyre, whose acoustical chamber is formed like a shell.
13. (music) The body of a drum; the often wooden, often cylindrical acoustic chamber, with or without rims added for tuning and for attaching the drum head.
14. An engraved copper roller used in print works.
15. (nautical) The watertight outer covering of the hull of a vessel, often made with planking or metal plating.
16. (nautical, rigging) The outer frame or case of a block within which the sheaves revolve.
17. (nautical) A light boat whose frame is covered with thin wood, impermeable fabric, or water-proofed paper; a racing shell or dragon boat.
18. (chemistry) A set of atomic orbitals that have the same principal quantum number.
19. (figuratively) The outward form independent of what's inside
20. (figratively) The empty outward form of someone or something.
21. An emaciated person.
22. A psychological barrier to social interaction.
23. (computing) An operating system software user interface, whose primary purpose is to launch other programs and control their interactions; the user's command interpreter.
24. (business) A legal entity that has no operations.
25. A concave rough cast-iron tool in which a convex lens is ground to shape.
26. (engineering) A gouge bit or shell bit.
27. (phonology) The onset and coda of a syllable.

Mathematics

Calculus

The Cylindrical Shell Method (Parallel Slices)

Used when slicing parallel to the rotation axis is simpler (e.g., when solving for \(x\) in terms of \(y\) is algebraically messy or impossible).

Unrolling a thin concentric cylindrical shell produces a flat rectangular slab of volume:

\[dV = (\text{circumference}) \times (\text{height}) \times (\text{thickness}) = 2\pi \cdot r \cdot h \cdot dx\]

• Rotation about a vertical axis (integrating with respect to \(x\)):

\[V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx = 2\pi \int_{a}^{b} x \cdot [f(x) - g(x)] \, dx\]

• Rotation about a horizontal axis (integrating with respect to \(y\)):

\[V = 2\pi \int_{c}^{d} y \cdot [f(y) - g(y)] \, dy\]