--- title: "Volumes of Revolution: Disks, Washers and Shells" description: "Set up a volume as an integral of cross-sections, and choose between slicing perpendicular to the axis and using cylindrical shells based on which integral is easier." canonical: https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4 source: https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Volumes of Revolution: Disks, Washers and Shells Set up a volume as an integral of cross-sections, and choose between slicing perpendicular to the axis and using cylindrical shells based on which integral is easier. Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4 ## Ready when they can - Write the integral for a solid generated by rotating a region about the x-axis - Use washers when the solid has a hole and identify both radii - Compare shell and disk or washer setups for the same solid, explaining which needs less rewriting or fewer pieces ## Lesson: Turning flat regions into volumes A volume integral adds cross-sectional area times slice thickness. Rotate 0 <= y <= f(x), a <= x <= b, about the x-axis, with f continuous and nonnegative. Vertical strips are perpendicular to the axis and make disks. Their radius is f(x), so V=π∫_a^b [f(x)]² dx. A thin slice contributes about π[f(x)]²Δx. If f(x)=c throughout, equal disks form a cylinder of radius c and length b-a. **Example.** Rotate 0 <= y <= √(2x), 0 <= x <= 3, about the x-axis. Squaring the radius gives 2x, so V=π∫_0^3 2x dx. An antiderivative of x is x²/2, so an antiderivative of 2x is x². Subtract endpoint values: V=π(3²-0²)=9π cubic units. A perpendicular strip that misses the axis makes a washer. Its outer radius R and inner radius r satisfy 0 <= r <= R. Subtract circle areas: A=π(R²-r²). A disk has r=0. Square each radius before subtracting; (R-r)²=R²-2Rr+r² is a different expression. For constant radii, volume is washer area times the interval length. With R=6 and r=2, the area is π(36-4)=32π. **Example.** Rotate the region between y=3x and y=x², for 0 <= x <= 3, about the x-axis. Both curves are nonnegative, and 3x>=x² on this interval. Thus R=3x and r=x². The volume setup is V=π∫_0^3 [(3x)²-(x²)²] dx=π∫_0^3 (9x²-x⁴) dx. Each radius is a distance from the axis, not the gap between the curves. Shells come from strips parallel to the axis. For a region on one side of the axis, a thin shell contributes about 2πrhΔt, where Δt is strip thickness. This is circumference times strip length times thickness. Add the contributions and take a limit. About the y-axis, vertical strips with x>=0 use r=x, h=top-bottom and V=2π∫ rh dx. About the x-axis, horizontal strips with y>=0 use r=y, h=right-left and V=2π∫ rh dy. **Example.** Rotate 0 <= y <= x³, 0 <= x <= 2, about the y-axis. Shells give V=2π∫_0^2 x⁴ dx=64π/5. Horizontal strips run from x=y^(1/3) to x=2, so they make washers: V=π∫_0^8 (4-y^(2/3)) dy=64π/5. Both methods describe the same solid. Shells keep x as the variable and avoid solving the curve for x; washers remain valid. **Example.** Rotate the triangle bounded by y=x, y=4-x and y=0 about the x-axis. Vertical disks need a split at x=2 because the upper boundary changes there. Horizontal shells need one integral. Solve the lines as x=y and x=4-y. For 0 <= y <= 2, radius is y and strip length is (4-y)-y=4-2y. Thus V=2π∫_0^2 y(4-2y) dy. Radii are nonnegative distances from the axis. About y=k, the distance from height y is |y-k|; about x=c, it is |x-c|. For a perpendicular strip on one side of the axis, compare its endpoint distances to choose R and r. Below the x-axis, y=-5 is farther away than y=-2, so R=5 and r=2. If a perpendicular strip crosses the axis, its slice is a full disk with radius equal to the greatest distance reached. *(drawing: Rotate 4 <= y <= 7, 0 <= x <= 3 about the x-axis. The washer area is constant, so multiply it by length 3.)* **Recap.** Measure radii from the axis. Perpendicular slices use π(R²-r²); parallel shells use 2πrh. Keep the limits and formulas in the same integration variable. *Lesson, practice questions and flashcards written for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Areas Between Two Curves](https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b) ## Optional connections - [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z) - [Volume and Surface Area of Cones, Spheres and Pyramids](https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-cones-spheres-and-pyramids-mt_s4K5YMBSzG) ## Opens up - [Arc Length and the Area of a Surface of Revolution](https://lightmysky.com/learn/mathematics/arc-length-and-the-area-of-a-surface-of-revolution-mt_W4uujT9Z1t)