{
  "id": "shell-method",
  "version": "fe15952c36dd",
  "status": "published",
  "name": "Shell Method Calculator",
  "question": "What is the shell method volume?",
  "summary": "Finds the volume of the solid made by turning the region between y = f(x) and y = g(x), a ≤ x ≤ b, about a vertical line x = k.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/shell-method-calculator",
  "markdown": "https://www.acalculator.org/math/shell-method-calculator.md",
  "kind": "function",
  "method": "V = 2π ∫ₐᵇ |x − k| |f(x) − g(x)| dx: shells of radius |x − k| and height |f − g|.",
  "assumptions": [
    "The axis x = k is outside the open interval (a, b).",
    "Angles in radians; an answer that fails its check is not shown."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Top curve y = f(x)",
        "description": "One edge of the region.",
        "type": "string",
        "maxLength": 200
      },
      "g": {
        "title": "Bottom curve y = g(x) (empty is 0)",
        "description": "The other edge; empty means the x-axis.",
        "type": "string",
        "maxLength": 200
      },
      "a": {
        "title": "From x = a",
        "description": "From x = a: a number or a constant such as pi.",
        "type": "string",
        "maxLength": 40
      },
      "b": {
        "title": "To x = b",
        "description": "To x = b: a number or a constant such as pi.",
        "type": "string",
        "maxLength": 40
      },
      "k": {
        "title": "Axis x = k",
        "description": "Axis x = k: a number or a constant such as pi.",
        "type": "string",
        "maxLength": 40
      }
    }
  },
  "outputs": {
    "volume": {
      "label": "Volume",
      "description": "2π ∫ |x − k| |f(x) − g(x)| dx from a to b.",
      "format": "number"
    },
    "exact": {
      "label": "Exact volume",
      "description": "The volume, exactly.",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "2x - x^2",
      "a": "0",
      "b": "2",
      "k": "0"
    },
    "outputs": {
      "volume": 8.377580409572781,
      "exact": "8π/3"
    },
    "text": "The shell method volume for y = 2x - x^2 from x = 0 to x = 2 about the line x = 0 is 8.37758041."
  },
  "examples": [
    {
      "given": {
        "f": "2x - x^2",
        "a": "0",
        "b": "2",
        "k": "0"
      },
      "expect": {
        "volume": 8.377580409572781,
        "exact": "8π/3"
      },
      "source": "OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells), Example 6.13 (about the y-axis)"
    },
    {
      "given": {
        "f": "x",
        "a": "1",
        "b": "2",
        "k": "-1"
      },
      "expect": {
        "volume": 24.085543677521745,
        "exact": "23π/3"
      },
      "source": "OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells), Example 6.15 (about the line x = −1)"
    },
    {
      "given": {
        "f": "sqrt(x)",
        "g": "1/x",
        "a": "1",
        "b": "4",
        "k": "0"
      },
      "expect": {
        "volume": 59.06194188748812,
        "exact": "94π/5"
      },
      "source": "OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells), Example 6.16 (between √x and 1/x, about the y-axis)"
    },
    {
      "given": {
        "f": "1/x",
        "a": "1",
        "b": "3",
        "k": "0"
      },
      "expect": {
        "volume": 12.566370614359172,
        "exact": "4π"
      },
      "source": "OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells), Example 6.12"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells"
  ],
  "related": [
    "washer-method",
    "area-between-curves",
    "integral",
    "volume"
  ],
  "changelog": []
}