Volumes of revolution – discs, washers, shells and cross-sections
MathematicsCalculusAges 17–18
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Sign in to playRotate the region between y = f(x) and y = g(x) about the x-axis, the y-axis or a line y = k or x = k to build a 3D solid of revolution. The solid is cut into n discs, washers or cylindrical shells whose total volume approaches V = π∫(R² − r²)dx as n grows, with a worked solution from antiderivatives. A cross-section mode builds square, semicircle and triangle slices on a base region.
Lesson: Applications of integration: volumes of solids of revolution and solids with known cross-sections
What it shows
A plane region bounded by y = f(x), y = g(x), x = a and x = b is rotated about an axis to form a solid. Slices perpendicular to the axis are discs or washers with volume π(R² − r²)Δx; slices parallel to a vertical axis are cylindrical shells with volume 2πρhΔx. As the number of slices n grows, their total approaches the definite integral, which the sim evaluates exactly from antiderivatives and writes as a multiple of π when possible. A second mode stacks square, semicircular or triangular cross-sections on the base region.
How to use
Pick f(x), g(x) and the limits a and b, then choose the Axis of rotation and, for a vertical axis, the Slicing method. Press Revolve region 360° to sweep the region into the solid, and drag Slice shown to read R, r and the volume of one slice. Increase n or press Run to watch Sₙ approach V. Try the Vase, Bowl, Sphere, Cone and Washer examples, or switch to Known cross-sections.
Parameters you can change
- Mode Solid of revolution, Known cross-sections
- View 3D, 2D (plane figure)
- Curve y = f(x) y = √x, y = x/2, y = x²/4, y = 1, y = 2, y = √(4 − (x − 2)²) (semicircle), y = 1.3 + 0.7·sin(0.4πx) (vase), y = sin x, y = e^(x/4)
- Curve y = g(x) y = 0 (x-axis), y = √x, y = x/2, y = x²/4, y = 1, y = 2, y = √(4 − (x − 2)²) (semicircle), y = 1.3 + 0.7·sin(0.4πx) (vase), y = sin x, y = e^(x/4)
- Lower limit a 0–4
- Upper limit b 0–4
- Axis of rotation x-axis (y = 0), Line y = k, y-axis (x = 0), Line x = k
- Position of the line k (y = k or x = k) -2–5
- Slicing method for a vertical axis Discs / washers (perpendicular to axis), Shells (parallel to axis)
- Cross-section shape Square of side s, Semicircle of diameter s, Equilateral triangle, side s, Isosceles right triangle, leg s
- Number of slices n 1–100
Questions to explore
- Rotating the region under y = x/2 on [0, 4] about the x-axis gives which cone, and does V match ⅓πr²h?
- When the region between y = x/2 and y = x²/4 is rotated about the x-axis, why must π∫r²dx be subtracted?
- For the bowl, do cylindrical shells and horizontal washers give the same volume?
