{
  "id": "triple-integral",
  "version": "0b3d7ed55a15",
  "status": "published",
  "name": "Triple Integral Calculator",
  "question": "What is the triple integral?",
  "summary": "Evaluates an iterated integral exactly where possible, checked numerically.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/triple-integral-calculator",
  "markdown": "https://www.acalculator.org/math/triple-integral-calculator.md",
  "kind": "function",
  "method": "Iterated CAS definite integrals, checked against tanh-sinh cubature.",
  "assumptions": [
    "Finite limits; 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": "Function f(x, y, z)",
        "description": "The integrand.",
        "type": "string",
        "maxLength": 200
      },
      "order": {
        "title": "Order",
        "description": "dz dy dx: inner in z, then y, outer in x.",
        "type": "string",
        "maxLength": 16
      },
      "a1": {
        "title": "Inner lower",
        "description": "The inner lower limit.",
        "type": "string",
        "maxLength": 60
      },
      "b1": {
        "title": "Inner upper",
        "description": "The inner upper limit.",
        "type": "string",
        "maxLength": 60
      },
      "a2": {
        "title": "Middle lower",
        "description": "The middle lower limit.",
        "type": "string",
        "maxLength": 60
      },
      "b2": {
        "title": "Middle upper",
        "description": "The middle upper limit.",
        "type": "string",
        "maxLength": 60
      },
      "a3": {
        "title": "Outer lower",
        "description": "The outer lower limit.",
        "type": "string",
        "maxLength": 60
      },
      "b3": {
        "title": "Outer upper",
        "description": "The outer upper limit.",
        "type": "string",
        "maxLength": 60
      }
    }
  },
  "outputs": {
    "exact": {
      "label": "Exact value",
      "description": "The integral, exact.",
      "format": "math"
    },
    "value": {
      "label": "Decimal value",
      "description": "The integral to 10 significant figures.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "5x - 3y",
      "order": "dz dy dx",
      "a1": "0",
      "b1": "1 - x - y",
      "a2": "0",
      "b2": "1 - x",
      "a3": "0",
      "b3": "1"
    },
    "outputs": {
      "exact": "1/12",
      "value": 0.08333333333333333
    },
    "text": "The triple integral of 5x - 3y (dz dy dx) is 0.08333333333."
  },
  "examples": [
    {
      "given": {
        "f": "5x - 3y",
        "order": "dz dy dx",
        "a1": "0",
        "b1": "1 - x - y",
        "a2": "0",
        "b2": "1 - x",
        "a3": "0",
        "b3": "1"
      },
      "expect": {
        "exact": "1/12",
        "value": 0.08333333333333333
      },
      "source": "OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 5.4, Ex. 5.38"
    },
    {
      "given": {
        "f": "1",
        "order": "dx dy dz",
        "a1": "-1 + z",
        "b1": "1 - z",
        "a2": "-1 + z",
        "b2": "1 - z",
        "a3": "0",
        "b3": "1"
      },
      "expect": {
        "exact": "4/3",
        "value": 1.3333333333333333
      },
      "source": "OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 5.4, Ex. 5.39"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 3, section 5.4 Triple Integrals: https://openstax.org/books/calculus-volume-3/pages/5-4-triple-integrals",
    "OpenStax, Calculus Volume 3, section 5.5 Triple Integrals in Cylindrical and Spherical Coordinates: https://openstax.org/books/calculus-volume-3/pages/5-5-triple-integrals-in-cylindrical-and-spherical-coordinates"
  ],
  "related": [
    "double-integral",
    "integral",
    "jacobian",
    "partial-derivative"
  ],
  "changelog": []
}