2a:null 2b:["$","title",null,{"children":"The Riemann integral | Mattias Villani | Observable"}] 31:T408,plt_func_riemann_static = Plot.plot({ x: { label: "x", axis: true, domain: xdomain, grid: false }, y: { label: "f(x)", axis: true, domain: ydomain, grid: false }, tooltip: { fill: "#3367b0", stroke: "#3367b0", opacity: 1, }, marks: [ Plot.ruleY([0]), Plot.ruleX([-5]), Plot.line([{x: a, y: ydomain[0]},{x: a, y: ydomain[1]}], {x: "x", y: "y", stroke: "orange"}), Plot.line([{x: b, y: ydomain[0]},{x: b, y: ydomain[1]}], {x: "x", y: "y", stroke: "orange"}), Plot.text([{x: a, y: ydomain[1]},{x: b, y: ydomain[1]}], {x: "x", y: "y", text: ['a','b'], dx: 0, dy: -7, fill: "orange"}), Plot.rectY(bindata, {filter: d => d.x1 >= a && d.x2 <= b, x1: "x1", x2: "x2", y: "y", strokeWidth:0 ,fill: "steelblue", opacity: 0.5, insetLeft: 0.444, title: (d) => `Rectangle area = ${(0.444*d.y).toPrecision(4)}`}), Plot.lineY(funcdata_dense,{x: "x", y: "y", stroke: "black"}) ] })32:T66f,The **Riemann rectangle sum** (area of blue rectangles) ${tex.block`\sum_{i=1}^n f(x_i^\star)\Delta x_i`} gets closer and closer (converges) to the Riemann integral ${tex`\int_a^b f(x)\mathrm{d}x`} as the grid becomes finer and finer, i.e. when ${tex`\Delta \rightarrow 0`}, if the function is smooth (integrable). This is (kind of) how the Riemann integral is **defined**, as the mathematical limit of the rectangle sum when ${tex`\Delta \rightarrow 0`}. The formal definition of a Riemann integral is actually based on two rectangle approximations, one where the top of the rectangles are always *above* the function ${tex`f(x)`} and one where the top of the rectangles are always *below* the function. The function is (Riemann) integrable if these two rectangle sums converge to the same value when the width (base) of the rectangles approaches zero. The integral sign ${tex`\int`} and differential symbol ${tex`\mathrm{d}x`} in the notation for an integral were chosen with care: - the integral sign ${tex`\int`} looks like the letter s for the word *sum*. - the differential symbol ${tex`\mathrm{d}x`} represents a really small version of ${tex`\Delta x`}, approaching zero (you can use the phrase *infinitesimally small* at cocktail parties 😊). Hopefully, the notation ${tex`\sum_{i=1}^n f(x_i^\star)\Delta x_i \rightarrow \int_a^b f(x)\mathrm{d}x`} as ${tex`\Delta x_i \rightarrow 0`} is starting to make sense now. You can experiment below to see that the Riemann sum really is close to the true integral when ${tex`\Delta`} is lowered to a value close to zero.33:T401,plt_func_riemann = Plot.plot({ x: { label: "x", axis: true, domain: xdomain, grid: false }, y: { label: "f(x)", axis: true, domain: ydomain, grid: false }, tooltip: { fill: "#3367b0", stroke: "#3367b0", opacity: 1, }, marks: [ Plot.ruleY([0]), Plot.ruleX([-5]), Plot.line([{x: a, y: ydomain[0]},{x: a, y: ydomain[1]}], {x: "x", y: "y", stroke: "orange"}), Plot.line([{x: b, y: ydomain[0]},{x: b, y: ydomain[1]}], {x: "x", y: "y", stroke: "orange"}), Plot.text([{x: a, y: ydomain[1]},{x: b, y: ydomain[1]}], {x: "x", y: "y", text: ['a','b'], dx: 0, dy: -7, fill: "orange"}), Plot.rectY(bindata, {filter: d => d.x1 >= a && d.x2 <= b, x1: "x1", x2: "x2", y: "y", strokeWidth:0 ,fill: "steelblue", opacity: 0.5, insetLeft: delta, title: (d) => `Rectangle area = ${(delta*d.y).toPrecision(4)}`}), Plot.lineY(funcdata_dense,{x: "x", y: "y", stroke: "black"}) ] })34:T66d,>

**Example**. Let us try to integrate the function ${tex`f(x)=x^${p_input.n}`} from ${tex`a=${p_input.a}`} to ${tex`b=${p_input.b}`}, see the figure below. The anti-derivative is the function whose derivative is ${tex`f(x)=x^${p_input.n}`}. We know that ${tex`\frac{\mathrm{d}}{\mathrm{d} x}x^${p_input.n+1}= ${p_input.n+1}x^${p_input.n}`}, so an anti-derivative to ${tex`f(x)=x^${p_input.n}`} is ${tex`\frac{1}{${p_input.n+1}}x^${p_input.n+1}.`} However, since additive constants disappear (have derivative zero) when taking the derivative, the function ${tex`\frac{1}{${p_input.n+1}}x^${p_input.n+1} + C`} for *any* constant ${tex`C`} is also an anti-derivative to ${tex`f(x)=x^${p_input.n}`}. The constant ${tex`C`} will cancel out when computing the definite integral, so we can safely ignore it here. Now the sought integral is ${tex.block`\int_{${p_input.a}}^{${p_input.b}} x^{${p_input.n}} \mathrm{d} x = \Big[\frac{1}{${p_input.n+1}}x^{${p_input.n+1}} \Big]_{${p_input.a}}^{${p_input.b}} =\frac{1}{${p_input.n +1}}(${p_input.b}^{${p_input.n+1}} - ${p_input.a}^{${p_input.n+1}}) \approx \frac{1}{${p_input.n +1}} (${(p_input.b**(p_input.n+1) - p_input.a**(p_input.n+1)).toFixed(3)}) \approx ${((p_input.b**(p_input.n+1)-p_input.a**(p_input.n+1))/(p_input.n +1)).toFixed(3)}`} > Note the convenience of using the bracket notation ${tex`[F(x)]_a^b`} with the anti-derivative inside the brackets. > Move the sliders to change the power in the function ${tex`f(x)`}, and the lower ${tex`a`} and upper ${tex`b`} limits of the integral. Note how all the numbers in the example box above also changes!35:[{"id":"a8b34f0b125309e2081a17b572774b256c23f9efdf233f3ab4bbb5b9e760a89dadb03ceea2ae18cd40a11db91a4fc3f061b70dd233b92391d7953abb3e612066","href":"https://static.observableusercontent.com/files/a8b34f0b125309e2081a17b572774b256c23f9efdf233f3ab4bbb5b9e760a89dadb03ceea2ae18cd40a11db91a4fc3f061b70dd233b92391d7953abb3e612066","name":"plt_func_riemann (2).svg","type":"image/svg+xml","lastModified":1738696460855.825,"size":21331}] 2e:{"id":"f4bcee1add8c851c","slug":"the-riemann-integral","title":"The Riemann integral","thumbnail":"0443671d50567f10436c21e048d295538fbc3e05ba613992b9d9be740b346d70","access_level":"public","create_time":"$D2023-11-14T22:22:59.495Z","update_time":"$D2025-03-16T10:39:36.500Z","view_time":null,"first_public_time":"$D2023-11-15T00:19:43.625Z","fork":{"id":"e6c40551335944ad","slug":"zipf-distribution","title":"Zipf distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":2479,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":3,"collection":{"id":"186f48bbffc13b97","slug":"stm","title":"STM","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"The Riemann integral","cells":[{"id":38,"value":"# The Riemann integral","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"The (definite) **integral** \n${tex.block`\\int_a^b f(x) \\mathrm{d}x`} \nof a function ${tex`f(x)`} measures the **area** under the function between ${tex`x=a`} and ${tex`x=b`}. Try with different functions:","pinned":false,"hidden":false,"mode":"md"},{"id":1871,"value":"viewof funcname = Inputs.select([\"linear\", \"exp\", \"log\", \"square\", \"cube\", \"sin\", \"gaussian\"], {value: \"square\", label: \"Function\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof inputs = Inputs.form([\n Inputs.range(limits, {\n value: 0,\n step: 0.1,\n label: tex`a`\n }),\n Inputs.range(limits, {\n value: 3,\n step: 0.1,\n label: tex`b`\n })\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":2037,"value":"[Riemann integral](https://en.wikipedia.org/wiki/Riemann_integral):\n\n${tex`\\int_a^b f(x) \\mathrm{d}x=\\int_{${a}}^{${b}}`} ${latexfunc} ${tex`\\mathrm{d}x = `} ${true_integral.toFixed(4)}\n\nTry changing *a* and *b* using the sliders to see how the integral (blue area) changes.","pinned":false,"hidden":false,"mode":"md"},{"id":1713,"value":"plt_func = Plot.plot({\n x: {\n label: \"x\",\n axis: true,\n domain: xdomain\n },\n y: {\n label: \"f(x)\",\n axis: true,\n domain: ydomain,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([-5]),\n Plot.line([{x: a, y: ydomain[0]},{x: a, y: ydomain[1]}], {x: \"x\", y: \"y\", stroke: \"orange\", label: \"a\"}),\n Plot.line([{x: b, y: ydomain[0]},{x: b, y: ydomain[1]}], {x: \"x\", y: \"y\", stroke: \"orange\", label: \"b\"}),\n Plot.text([{x: a, y: ydomain[1]},{x: b, y: ydomain[1]}], {x: \"x\", y: \"y\", text: ['a','b'], dx: 0, dy: -7, fill: \"orange\"}), \n Plot.lineY(funcdata_dense,{x: \"x\", y: \"y\", stroke: \"black\"}),\n Plot.areaY(funcdata_dense, {filter: d => d.x >= a && d.x <= b, x: \"x\", y: \"y\", fill: \"steelblue\", opacity: 0.5}),\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":2218,"value":"Integration is essential in Statistics to compute probabilities\n${tex.block`\\mathrm{Pr}(a\\leq X \\leq b)`} \nfor some random variable ${tex`X`}. For *discrete random variables* such probabilities can be computed with simple sums. Try below with a discrete random variable with support ${tex`\\{0,1,2,3,4,5\\}`}. The probability ${tex`\\mathrm{Pr}(a\\leq X \\leq b)`} is the sum of the height of the dark blue bars.","pinned":false,"hidden":false,"mode":"md"},{"id":2266,"value":"viewof problimits = Inputs.form({\n a: Inputs.range([0, 5], {value: 2, label: \"a\", step: 1}),\n b: Inputs.range([0, 5], {value: 3, label: \"b\", step: 1})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":2279,"value":"probtext = tex`\n\\mathrm{Pr}( ${problimits.a} \\leq X \\leq ${problimits.b}) = \\sum_{k=${problimits.a}}^${problimits.b} \\mathrm{Pr}(X=k) = ${prob_discrete_interval.toPrecision(4)}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2248,"value":"plt_pdf = Plot.plot({\n x: {\n label: \"x\",\n axis: true,\n domain: d3.range(0, 6, 1)\n },\n y: {\n label: \"P(x)\",\n //domain: [0,1.1*jstat.binomial.pdf(params[0]*params[1], params[0],params[1])]\n },\n tooltip: {\n fill: \"#0F52BA\",\n stroke: \"#0F52BA\",\n opacity: 1,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.barY(pdfdata,{x: \"x\", y: \"pdf\", fill : \"#B6D0E2\", strokeWidth: 0, opacity: 0.8, \n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n Plot.barY(pdfdata, {filter: d => d.x >= problimits.a && d.x <= problimits.b, x: \"x\", y: \"pdf\", fill: \"#6F8FAF\", opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":2291,"value":"For *continuous random variables* we compute probabilities as the **area** under a *probability density function* (pdf), ${tex`f(x)`}. For example, the pdf of a standard Gaussian random variable is of the form\n${tex.block`f(x)=\\frac{1}{\\sqrt{2\\pi}}\\exp\\Big(-\\frac{1}{2}x^2\\Big)`} \nWe can then use integration to compute the probability as the area under the pdf between ${tex`a`} and ${tex`b`}: \n${tex.block`\\mathrm{Pr}(a\\leq X \\leq b) = \\int_a^b f(x)\\mathrm{d}x`}\nYou can try this by choosing *gaussian* in the drop-down menu in the plot at the top of the notebook. Note that this density function is never negative (which is true for any pdf) and integrates to one over its support, ${tex`\\mathrm{Pr}(-\\infty \\leq X \\leq \\infty) = \\int_a^b f(x)\\mathrm{d}x = 1`}, (also true for any pdf). ","pinned":false,"hidden":false,"mode":"md"},{"id":2058,"value":"For functions ${tex`f(x)`} that can be negative, for example ${tex`\\sin(x)`}, the integral can be negative 🥴. It may seem a little strange to have a negative area, but that is how the Riemann integral is defined.","pinned":false,"hidden":false,"mode":"md"},{"id":2193,"value":"The [Riemann integral](https://en.wikipedia.org/wiki/Riemann_integral) of the function ${tex`f(x)`} is based on a **rectangle approximation** of the area under ${tex`f(x)`} between ${tex`x=a`} and ${tex`x=b`}. \n
The figure below shows a rather bad approximation with too wide rectangle bases; we will soon do better, using thinner and thinner rectangles.","pinned":false,"hidden":false,"mode":"md"},{"id":2188,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":1708,"value":"The mathematical formulation of the rectangle approximation of the area under ${tex`f(x)`} between ${tex`x=a`} and ${tex`x=b`} is\n\n${tex.block`\\sum_{i=1}^n f(x_i^\\star)\\Delta x_i`}\nwhere ${tex`\\Delta x_i = x_i-x_{i-1}`} is the base of the *i*th rectangle and\n${tex.block`x_0 = a < x_1 < x_2 < \\ldots < x_{n-1} < x_n=b`} \n\nis a **grid** of *x*-values between *a* and *b*, and ${tex`x_i^\\star`} is the midpoint between the two grid points ${tex`x_{i-1}`} and ${tex`x_i`}. We will here use equally spaced grid points so that the distance is always the same number ${tex`\\Delta`}, but this is not required for the Riemann integral.","pinned":false,"hidden":false,"mode":"md"},{"id":1673,"value":"$32","pinned":false,"hidden":false,"mode":"md"},{"id":2087,"value":"anchor = Inputs.bind(\n Inputs.select([\"linear\", \"exp\", \"log\", \"square\", \"cube\", \"sin\", \"gaussian\"], {value: \"square\", label: \"Function\"})\n ,viewof funcname\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1751,"value":"Inputs.bind(\n Inputs.form([\n Inputs.range(limits, {\n value: -3,\n step: 0.1,\n label: tex`a`\n }),\n Inputs.range(limits, {\n value: 1,\n step: 0.1,\n label: tex`b`\n })\n ]),\n viewof inputs\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1738,"value":"viewof delta = Inputs.range([0.0005, 1], {\n value: 0.75,\n step: 0.0005,\n label: tex`\\text{bar width }\\Delta`\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":1840,"value":"$${tex`\\sum_{i=1}^n f(x_i^\\star)\\Delta x_i`} = ${Riemann_sum.toFixed(4)}\n${tex`\\quad \\rightarrow \\quad \\int_a^b f(x)dx = ${true_integral.toFixed(4)}\\quad \\text{as} \\quad \\Delta = ${delta.toFixed(4)} \\rightarrow 0`}

\n\n","pinned":false,"hidden":false,"mode":"md"},{"id":305,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":2341,"value":"It would be a nightmare is we had to take the limit of the Riemann sum every time we want to integrate a function. Luckily there is a much simpler route using something called the anti-derivative of a function.
","pinned":false,"hidden":false,"mode":"md"},{"id":2452,"value":">

**Definition (anti-derivative)**: A function ${tex`F(x)`} is the anti-derivative to the function ${tex`f(x)`} if\n${tex.block`F'(x)=f(x),\\text{ for all }x`}","pinned":false,"hidden":false,"mode":"md"},{"id":2348,"value":"Anti-derivatives are life-savers when it comes to integration since they can be used to compute definite integrals\n${tex.block`\\int_a^b f(x) \\mathrm{d} x = [F(x)]_a^b = F(b) - F(a),`}\nwhere ${tex`[F(x)]_a^b`} really just means ${tex`F(b) - F(a)`}, but is convenient notation when doing the integration for a specific function, as we will see in the following example.","pinned":false,"hidden":false,"mode":"md"},{"id":2356,"value":"$34","pinned":false,"hidden":false,"mode":"md"},{"id":2382,"value":"viewof p_input = Inputs.form({\n n: Inputs.range([1, 5], {value: 2, label: \"power, n\", step: 1}),\n a: Inputs.range([-1, 5], {value: 1, label: \"lower limit, a\", step: 0.1}),\n b: Inputs.range([0, 5], {value: 3, label: \"upper limit, b\", step: 0.1}),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":2374,"value":"plt_func2 = Plot.plot({\n x: {\n label: \"x\",\n axis: true,\n domain: xdomain_power\n },\n y: {\n label: \"f(x)\",\n axis: true,\n domain: ydomain_power,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([-1]),\n Plot.line([{x: p_input.a, y: ydomain_power[0]},{x: p_input.a, y: ydomain_power[1]}], \n {x: \"x\", y: \"y\", stroke: \"orange\", label: \"a\"}),\n Plot.line([{x: p_input.b, y: ydomain_power[0]},{x: p_input.b, y: ydomain_power[1]}], \n {x: \"x\", y: \"y\", stroke: \"orange\", label: \"b\"}),\n Plot.text([{x: p_input.a, y: ydomain_power[1]},{x: p_input.b, y: ydomain_power[1]}], \n {x: \"x\", y: \"y\", text: ['a','b'], dx: 0, dy: -7, fill: \"orange\"}), \n Plot.lineY(funcdatasquare,{x: \"x\", y: \"y\", stroke: \"black\"}),\n Plot.areaY(funcdatasquare, {filter: d => d.x >= p_input.a && d.x <= p_input.b, \n x: \"x\", y: \"y\", fill: \"steelblue\", opacity: 0.5}),\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":2444,"value":"Here are some useful rules for integrals:","pinned":false,"hidden":false,"mode":"md"},{"id":2316,"value":"diff_rules = html`

\n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n

Integral rules

${tex`\\int x^n \\mathrm{d} x = \\frac{1}{n+1} x^{n+1} + C \\text{ for } n \\neq -1`}

${tex`\\int e^{ax} \\mathrm{d} x = \\frac{1}{a} e^{ax} + C \\text{ for } a \\neq 0`}

${tex`\\int \\frac{1}{x} \\mathrm{d} x = \\ln\\vert x \\vert + C \\text{ for } x>0`}

${tex`\\int k f(x) \\mathrm{d} x = k \\int f(x) \\mathrm{d} x \\text{ for constant } k`}

${tex`\\int (f(x) + g(x)) \\mathrm{d} x = \\int f(x)\\mathrm{d} x + \\int g(x) \\mathrm{d} x `}

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is some code and data details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":1375,"value":"import {Plot} from \"@mkfreeman/plot-tooltip\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":2318,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":750,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":654,"value":"a = inputs[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":656,"value":"b = inputs[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1877,"value":"f = {\n if (funcname == \"linear\"){\n return function(x){return x}\n }else if(funcname == \"exp\"){\n return function(x){return Math.exp(x)}\n }else if(funcname == \"log\"){\n return function(x){return Math.log(x)}\n }else if(funcname == \"square\"){\n return function(x){return x**2}\n }else if(funcname == \"cube\"){\n return function(x){return x**3}\n }else if(funcname == \"gaussian\"){\n return function(x){return (1/Math.sqrt(2*Math.PI))*Math.exp(-0.5*x**2)}\n }else{\n return function(x){return Math.sin(x)}\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1975,"value":"latexfunc = {\n if (funcname == \"linear\"){\n return tex`x`\n }else if(funcname == \"exp\"){\n return tex`\\exp(x)`\n }else if(funcname == \"log\"){\n return tex`\\log(x)`\n }else if(funcname == \"square\"){\n return tex`x^2`\n }else if(funcname == \"cube\"){\n return tex`x^3`\n }else if(funcname == \"gaussian\"){\n return tex`\\frac{1}{\\sqrt{2\\pi}}\\exp(-\\frac{1}{2}x^2)`\n }else{\n return tex`\\sin(x)`\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1682,"value":"true_integral = {\n if (funcname == \"linear\"){\n return (b**2 - a**2)/2\n }else if(funcname == \"exp\"){\n return Math.exp(b) - Math.exp(a)\n }else if(funcname == \"log\"){\n return b*(Math.log(b)-1) - a*(Math.log(a)-1)\n }else if(funcname == \"square\"){\n return (b**3 - a**3)/3\n }else if(funcname == \"cube\"){\n return (b**4 - a**4)/4\n }else if(funcname == \"gaussian\"){\n return jstat.normal.cdf(b,0,1) - jstat.normal.cdf(a,0,1) \n }else{\n return Math.cos(a) - Math.cos(b)\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1199,"value":"gridsize = 0.01","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"funcdata = {\n return d3.range(xdomain[0], xdomain[1], delta).map(d => ({x: d, y: f(d)}))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1746,"value":"funcdata_dense = {\n return d3.range(xdomain[0], xdomain[1], 0.01).map(d => ({x: d, y: f(d)}))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1600,"value":"bindata = funcdata.map(d => ({x1: d.x, x2: d.x + delta, y: f(d.x + delta/2)}))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1628,"value":"Riemann_sum = d3.sum(funcdata.filter(d => d.x >= a && d.x <= b).map(d => d.y))*delta","pinned":false,"hidden":false,"mode":"ojs"},{"id":2080,"value":"xdomain = {\n if (funcname == \"log\"){\n return [1,5]\n }\n if (funcname == \"exp\"){\n return [-2,5]\n }\n return [-5,5]\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1929,"value":"ydomain = {\n if (funcname == \"gaussian\"){\n return [0,0.5]\n }else{\n return [Math.floor(d3.min(funcdata_dense.map(d => d.y))),Math.ceil(d3.max(funcdata_dense.map(d => d.y)))]\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1935,"value":"yrange = ydomain[1] - ydomain[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":2121,"value":"limits = [xdomain[0]+0.5,xdomain[1]+0.5]","pinned":false,"hidden":false,"mode":"ojs"},{"id":2250,"value":"pdfdata = {\n const x = d3.range(0, 5, 1);\n const pdf = x.map(x => ({x: x, pdf: 1/(2.5+x**2)}));\n return pdf.concat({x:5, pdf: 1-d3.sum(pdf.map(d => d.pdf))})\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":2378,"value":"funcdatasquare = {\n return d3.range(xdomain_power[0], xdomain_power[1], 0.01).map(d => ({x: d, y: d**p_input.n}))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":2394,"value":"xdomain_power = [-1,5]","pinned":true,"hidden":false,"mode":"ojs"},{"id":2388,"value":"ydomain_power = [Math.floor(d3.min(funcdatasquare.map(d => d.y))),Math.ceil(d3.max(funcdatasquare.map(d => d.y)))]","pinned":true,"hidden":false,"mode":"ojs"},{"id":2281,"value":"prob_discrete_interval = d3.sum(pdfdata.filter(d => d.x >= problimits.a && d.x <= problimits.b).map(d => d.pdf))","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W35","stdlib":"1","titling":"auto"}} 2f:"$2e:body"