1 · The disc method — spinning a region about the x-axis
Rotate the region under y = f(x), a ≤ x ≤ b, about the x-axis. Every vertical slice sweeps a thin disc of radius f(x) and thickness dx, with volume πf(x)² dx. Total: V = π ∫ₐᵇ f(x)² dx. For rotation about the y-axis of a region given as x = g(y), do the same with y: V = π ∫ g(y)² dy.
Step 1: disc radius = √x ⇒ disc area = π(√x)² = πx.
Step 2: V = π ∫₀⁴ x dx = π · [x²/2]₀⁴ = 8π.
Step 3: sanity check — paraboloid-like solid, volume slightly less than π·4²·4/2 = 32π. ✓
2 · Washers — rotation with a hole
When the region sits between an outer curve R(x) and an inner curve r(x) (both above the axis), slices sweep out washers: annulus area π(R(x)² − r(x)²). Volume: V = π ∫ₐᵇ (R(x)² − r(x)²) dx. Rotating about a horizontal line y = k ≠ 0? Shift the radii: use R − k and r − k in place of R and r.
3 · The shell method — cylindrical shells
Rotate vertical strips around the y-axis (or any y-axis-parallel line). A strip at x with height f(x) and thickness dx sweeps a shell of radius x, height f(x): volume 2π·x·f(x)·dx. Total: V = 2π ∫ₐᵇ x f(x) dx. Shells shine when the disc setup forces you to invert y = f(x) — rotating a region described by y = f(x) around the y-axis is the signature shells problem.
Step 1: shell radius x, shell height 2x − x², thickness dx.
Step 2: V = 2π ∫₀² x(2x − x²) dx = 2π ∫₀² (2x² − x³) dx = 2π [2x³/3 − x⁴/4]₀² = 2π(16/3 − 4) = 8π/3.
Step 3: the disc method would need x = 1 ± √(1 − y), messy square roots — shells win cleanly.
4 · Which method when — the summary
- About the x-axis, region y = f(x): discs/washers, integrate in x.
- About the y-axis, region y = f(x): shells, integrate in x (usually easiest).
- About the y-axis, region x = g(y): discs/washers, integrate in y.
- About the x-axis, region x = g(y): shells, integrate in y.
- Whole region down to the axis → disc. Region with a hole/between two curves → washer.
5 · Volumes of general solids via cross-sections
Slice ANY solid perpendicular to an axis with known cross-sectional area A(x): V = ∫ₐᵇ A(x) dx. Squares of side s(x): A = s². Equilateral triangles: A = (√3/4)s². Semicircles of diameter s: A = (π/8)s². E.g. the base region between y = x² and y = 1, with square cross-sections perpendicular to the y-axis: at height y the slice runs from x = −√y to x = +√y, so the square has side 2√y and area A(y) = 4y, giving V = ∫₀¹ 4y dy = 2.
6 · Arc length and surface area
Arc length: a small step dx corresponds to ds = √(1 + y′²) dx, so L = ∫ₐᵇ √(1 + (dy/dx)²) dx. For parametric curves, use speed: L = ∫ √((dx/dt)² + (dy/dt)²) dt. Surface area of a solid of revolution: S = 2π ∫ (radius) · ds = 2π ∫ₐᵇ f(x)√(1 + y′²) dx for rotation about the x-axis.
7 · Axis-parallel variations
Rotation about y = k (a line parallel to the x-axis): radius becomes |f(x) − k|. About x = h: shell radius becomes |x − h|, and washer setups need g(y) − h. The rule of thumb: always measure the radius from the axis of rotation, never blindly from the coordinate axes — draw the strip, label its radius, then integrate.
Shells: V = 2π ∫ (radius)(height) dx · Cross-sections: V = ∫ A(x) dx
Arc length: L = ∫ √(1 + y′²) dx = ∫ √((x′)² + (y′)²) dt
Surface: S = 2π ∫ f(x)√(1 + y′²) dx (about x-axis)
Parallel axis y = k: radius = |f(x) − k|
Common mistakes
- Writing π(R − r)² instead of π(R² − r²) for washers.
- Dropping the 2π in the shell formula — that gives πx·f(x), not a shell.
- Using vertical strips for shells about the x-axis — shells need strips parallel to the axis.
- Reading 1 + y instead of 1 + y′² under the arc-length radical.
- In surface area, integrating f(x) instead of f(x)·ds.