# Chapter 29: Volumes, Arc Lengths, and Surface Areas > Discs, washers, shells and cross-sections for volumes of revolution, plus arc length and surface area — the full toolkit for turning flat regions into exact 3D measurements. **Canonical:** https://calculus.kitzo.work/chapter/applications-integrals **Level:** Advanced **Estimated time:** 55 minutes **Exercises:** 8 (auto-marked; answers stay server-side) ## Topics - Disc method - Shell method - Cross-sections - Arc length - Parametric arc length - Surface area of revolution **Why this matters** Integrals do more than measure area — they accumulate anything that adds up uniformly: volume, arc length, surface area, mass, work. The disc, washer and shell methods are how you turn a flat 2D region into exact measurements of a 3D solid. ## 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. **Worked example** **Example 1 — Region under y = √x, 0 ≤ x ≤ 4, rotated about the x-axis.** 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. **Example 2 — Region bounded by y = x(2 − x) and y = 0, rotated about the y-axis.** 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. **Watch out** The √(1 + y′²) radical usually only cooperates when 1 + y′² is a perfect square — textbook curves are designed that way (y = (2/3)x^(3/2), since y′ = √x). And in surface area the radius is the distance to the axis — not the disc radius squared — missing the ds factor is the classic disaster. ## 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. **Chapter formula sheet** Discs: V = π ∫ f(x)² dx · Washers: V = π ∫ (R² − r²) dx 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. ## Practice exercises 1. The region under y = √x on [0, 4] is rotated about the x-axis. The volume is: - A. 8π - B. 4π - C. 16π - D. 2π 2. The region under y = x on [0, 2] rotates about the x-axis. Find the volume (decimal). 3. Find the arc length of y = (2/3)x^(3/2) from x = 0 to x = 3. 4. Rotating a region given as y = f(x) about the y-axis is usually easiest with the: - A. disc method - B. shell method - C. washer method - D. cross-section method 5. A solid is built on a region in the xy-plane with cross-sections perpendicular to the x-axis that are squares of side f(x) . Its cross-sectional area A(x) is (write like f(x)^2 ): 6. Find the surface area generated by rotating y = x on [0, 1] about the x-axis. 7. For a washer with outer radius R and inner radius r, the cross-sectional area is: - A. π(R + r) - B. π(R² − r²) - C. πR² − r - D. 2πRr 8. When using the shell method with vertical strips to rotate about the y-axis, the radius of the shell at position x is … (one letter). --- Practice interactively: https://calculus.kitzo.work/chapter/applications-integrals Citation: CalculusKitzo (https://calculus.kitzo.work/). Independent study notes inspired by the standard calculus syllabus; not affiliated with Princeton University Press. See https://calculus.kitzo.work/disclaimer