{
  "id": "washer-method",
  "version": "7c499b47122f",
  "status": "published",
  "name": "Washer Method Calculator",
  "question": "What is the washer 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 horizontal line y = k, by washers or disks.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/washer-method-calculator",
  "markdown": "https://www.acalculator.org/math/washer-method-calculator.md",
  "kind": "function",
  "method": "V = π ∫ₐᵇ |(f(x) − k)² − (g(x) − k)²| dx: washers with outer and inner radii |f − k| and |g − k|.",
  "assumptions": [
    "The region lies on one side of the axis y = k.",
    "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": "Outer curve y = f(x)",
        "description": "One edge of the region.",
        "type": "string",
        "maxLength": 200
      },
      "g": {
        "title": "Inner 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 y = k",
        "description": "Axis y = k: a number or a constant such as pi.",
        "type": "string",
        "maxLength": 40
      }
    }
  },
  "outputs": {
    "volume": {
      "label": "Volume",
      "description": "π ∫ |(f(x) − k)² − (g(x) − k)²| dx from a to b.",
      "format": "number"
    },
    "exact": {
      "label": "Exact volume",
      "description": "The volume, exactly.",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "x",
      "g": "1/x",
      "a": "1",
      "b": "4",
      "k": "0"
    },
    "outputs": {
      "volume": 63.61725123519331,
      "exact": "81π/4"
    },
    "text": "The washer method volume for y = x from x = 1 to x = 4 about the line y = 0 is 63.61725124."
  },
  "examples": [
    {
      "given": {
        "f": "x",
        "g": "1/x",
        "a": "1",
        "b": "4",
        "k": "0"
      },
      "expect": {
        "volume": 63.61725123519331,
        "exact": "81π/4"
      },
      "source": "OpenStax, Calculus Volume 1, section 6.2 Determining Volumes by Slicing (https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing), Example 6.10 (washers about the x-axis; the integral set up there, π∫₁⁴ (x² − 1/x²) dx, has f(x) = x)"
    },
    {
      "given": {
        "f": "4 - x",
        "a": "0",
        "b": "4",
        "k": "-2"
      },
      "expect": {
        "volume": 167.55160819145564,
        "exact": "160π/3"
      },
      "source": "OpenStax, Calculus Volume 1, section 6.2 Determining Volumes by Slicing (https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing), Example 6.11 (about the line y = −2)"
    },
    {
      "given": {
        "f": "sqrt(x)",
        "a": "1",
        "b": "4",
        "k": "0"
      },
      "expect": {
        "volume": 23.561944901923447,
        "exact": "15π/2"
      },
      "source": "OpenStax, Calculus Volume 1, section 6.2 Determining Volumes by Slicing (https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing), Example 6.8 (disks)"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 1, section 6.2 Determining Volumes by Slicing (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing"
  ],
  "related": [
    "shell-method",
    "area-between-curves",
    "integral",
    "volume"
  ],
  "changelog": []
}