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  "body": "\n## Definition\n\nLet $R = [a, b] \\times [c, d]$ with $a \u003c b$ and $c \u003c d$, and let $f \\colon R \\to \\mathbb{R}$ be bounded.\nChoose partitions $a = x_0 \u003c \\dots \u003c x_k = b$ and $c = y_0 \u003c \\dots \u003c y_l = d$, and let $R_{ij} = [x_{i-1}, x_i] \\times [y_{j-1}, y_j]$ with area $\\Delta A_{ij} = (x_i - x_{i-1})(y_j - y_{j-1})$.\nFor points $p_{ij} \\in R_{ij}$, the sum $S = \\sum_{i, j} f(p_{ij}) \\, \\Delta A_{ij}$ is a Riemann sum of $f$.\nThe function $f$ is Riemann integrable on $R$ if there is $I \\in \\mathbb{R}$ with the following property. For each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that $|S - I| \u003c \\varepsilon$ for all such sums whose rectangles have diameter less than $\\delta$.\nThen $I$ is the double integral of $f$ over $R$, written $\\iint_R f(x, y) \\, dA$ or $\\iint_R f(x, y) \\, dx \\, dy$.\n\n## Remarks\n\nEach [[continuous|0S9EK5MQPE]] function on $R$ is Riemann integrable on $R$.\nFor a bounded region $D \\subseteq R$, one integrates $f \\mathbf{1}_D$ over $R$, where $\\mathbf{1}_D$ is $1$ on $D$ and $0$ elsewhere. This gives [[integrals over non-rectangular regions|0TXCQ9CST1]].\nIn practice one computes double integrals as [[iterated integrals|0P4WX7Y502]], by [[Fubini's theorem|0R0T1VWH5X]]. For example, $\\iint_{[0, 1]^2} x y \\, dA = \\int_0^1 \\int_0^1 x y \\, dy \\, dx = 1/4$.\nThe integral is linear in $f$, and $\\iint_R 1 \\, dA = (b - a)(d - c)$ is the area of $R$.\n\n## Explanations\n\n### L1 human\n\nCut the rectangle into small pieces, multiply the area of each piece by a value of $f$ in it, and add.\nAs the pieces get smaller, these sums approach the double integral.\nFor $f \\ge 0$ this is the volume under the graph of $f$.\n",
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