{
  "aliases": [
    "area-by-greens-theorem"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $C$ be a positively oriented, piecewise smooth, simple [[closed curve|0D032NSZKY]] in $\\mathbb{R}^2$ that encloses the bounded region $D$.\nThen the area $A$ of $D$ satisfies\n$$A = \\oint_C x \\, dy = -\\oint_C y \\, dx = \\frac{1}{2} \\oint_C (x \\, dy - y \\, dx).$$\n\n## Remarks\n\nProof: apply [[Green's theorem|0HR3N02317]] with $(L, M)$ equal to $(0, x)$, $(-y, 0)$ and $(-y/2, x/2)$. In each case $\\partial M / \\partial x - \\partial L / \\partial y = 1$.\nExample: the ellipse $x = a \\cos t$, $y = b \\sin t$, $t \\in [0, 2\\pi]$, with $a, b \u003e 0$, gives $A = \\tfrac{1}{2} \\int_0^{2\\pi} a b \\, dt = \\pi a b$.\nFor a polygon with vertices $(x_1, y_1), \\dots, (x_n, y_n)$ in counterclockwise order, the formula gives $A = \\tfrac{1}{2} \\sum_{k=1}^n (x_k y_{k+1} - x_{k+1} y_k)$, with $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.\n\n## Explanations\n\n### L1 human\n\nYou can find the area of a region by going once around its boundary, without looking inside.\nThis is a direct use of Green's theorem with a field whose curl is $1$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "area-by-greens-theorem"
    ],
    "kind": "statement",
    "msc": [
      "26B20"
    ],
    "name": "Area of a region as a line integral",
    "role": "corollary",
    "sources": [
      {
        "locator": "Area calculation",
        "url": "https://en.wikipedia.org/wiki/Green%27s_theorem"
      }
    ],
    "tag": "09CBQB6V67",
    "uses": [
      "0HR3N02317",
      "0YS8RX87F4",
      "0D032NSZKY"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/09CBQB6V67/",
  "links": [
    {
      "field": "uses",
      "target": "0HR3N02317"
    },
    {
      "field": "uses",
      "target": "0YS8RX87F4"
    },
    {
      "field": "uses",
      "target": "0D032NSZKY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/09CBQB6V67.md",
  "msc": [
    "26B20"
  ],
  "name": "Area of a region as a line integral",
  "prov": null,
  "role": "corollary",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/09C/09CBQB6V67.md",
  "status": null,
  "tag": "09CBQB6V67",
  "uri": "m2://09CBQB6V67",
  "url": "https://math2.tamnd.com/wiki/area-of-a-region-as-a-line-integral/",
  "writing": {
    "tag": "09CBQB6V67",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}