// Numbas version: exam_results_page_options
{"name": "Integration: Calculating the area under a curve", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: Calculating the area under a curve", "tags": [], "metadata": {"description": "<p>Two quadratic graphs are sketched with some area beneath them shaded. Question is to determine the area of shaded regions using integration. The first graph's area is all above the $x$-axis. The second graph has some area above and some below the $x$-axis.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>Main advice is to carefully go through the steps given in lectures: make an estimate, establish the integral(s) required, do the calculations, check your answer.</p>\n<p>See&nbsp;??</p>", "rulesets": {}, "extensions": ["jsxgraph"], "variables": {"x21": {"name": "x21", "group": "b) quadratic. no neg region", "definition": "random(-4..-2)", "description": "", "templateType": "anything"}, "x31": {"name": "x31", "group": "c quadratic. neg region", "definition": "b3-random(2..3)", "description": "", "templateType": "anything"}, "c4": {"name": "c4", "group": "Ungrouped variables", "definition": "b4+2", "description": "", "templateType": "anything"}, "x32": {"name": "x32", "group": "c quadratic. neg region", "definition": "b3", "description": "", "templateType": "anything"}, "x33": {"name": "x33", "group": "c quadratic. neg region", "definition": "x32+2", "description": "", "templateType": "anything"}, "b3": {"name": "b3", "group": "c quadratic. neg region", "definition": "a3+random(3..4)", "description": "", "templateType": "anything"}, "area3": {"name": "area3", "group": "c quadratic. neg region", "definition": "(x33^3/3 - 0.5*(a3+b3)*x33^2+a3*b3*x33)-2*(x32^3/3 - 0.5*(a3+b3)*x32^2+a3*b3*x32)+(x31^3/3 - 0.5*(a3+b3)*x31^2+a3*b3*x31)", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "b) quadratic. no neg region", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(-3..-2)", "description": "", "templateType": "anything"}, "x22": {"name": "x22", "group": "b) quadratic. no neg region", "definition": "random(3..5)+x21", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "linear graph (not used)", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "c quadratic. neg region", "definition": "random(-3..-1)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "linear graph (not used)", "definition": "1", "description": "", "templateType": "anything"}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "a4+random(2..3)", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "b) quadratic. no neg region", "definition": "1", "description": "", "templateType": "anything"}, "x11": {"name": "x11", "group": "linear graph (not used)", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "area2": {"name": "area2", "group": "b) quadratic. no neg region", "definition": "(a2*x22^3/3+c2*x22)-(a2*x21^3/3+c2*x21)", "description": "", "templateType": "anything"}, "x12": {"name": "x12", "group": "linear graph (not used)", "definition": "random(1..4) + x11", "description": "", "templateType": "anything"}, "x41": {"name": "x41", "group": "Ungrouped variables", "definition": "a4", "description": "", "templateType": "anything"}, "x42": {"name": "x42", "group": "Ungrouped variables", "definition": "b4", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a4", "b4", "c4", "x41", "x42"], "variable_groups": [{"name": "linear graph (not used)", "variables": ["x11", "x12", "a1", "b1"]}, {"name": "b) quadratic. no neg region", "variables": ["x21", "x22", "a2", "c2", "area2"]}, {"name": "c quadratic. neg region", "variables": ["a3", "b3", "x31", "x32", "x33", "area3"]}], "functions": {"plotgraph": {"parameters": [["q", "number"], ["x1", "number"], ["x2", "number"], ["ymin", "number"], ["ymax", "number"], ["a", "number"], ["b", "number"], ["c", "number"]], "type": "html", "language": "javascript", "definition": "// Shading under a graph! This functions plots a graph of y = a(x-r1)(x-r2)\n// It creates the board, sets it up, then returns an\n// HTML div tag containing the board.\n\n\n// Max and min x and y values for the axis.\nvar xmin = -7;\nvar xmax = 7;\n\n// First, make the JSXGraph board.\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n  '500px',\n  '500px',\n {\n   boundingBox: [xmin,ymax,xmax,ymin],\n   axis: false,\n   showNavigation: false,\n   grid: true\n }\n);\n\n\n\n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar brd = div.board; \n\n// create the x-axis.\nvar xaxis = brd.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = brd.create('ticks',[xaxis,1],{\n  drawLabels: true,\n  label: {offset: [-4, -10]},\n  minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = brd.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nyticks = brd.create('ticks',[yaxis,5],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 4\n});\n\n\n\n// This function shades in the area below the graph of f\n// between the x values x1 and x2\n\nvar shade = function(f,x1,x2,colour) {\n  var dataX1 = [x1,x1];\n  var dataY1 = [0,f(x1)];\n\n  var dataX2 = [];\n  var dataY2 = [];\n  for (var i = x1; i <= x2; i = i+0.1) {\n      dataX2.push(i);\n      dataY2.push(f(i));\n  }\n\n  var dataX3 = [x2,x2];\n  var dataY3 = [f(x2),0];\n\n  dataX = dataX1.concat(dataX2).concat(dataX3);\n  dataY = dataY1.concat(dataY2).concat(dataY3);\n\nvar shading =  brd.create('curve', [dataX,dataY],{strokeWidth:0, fillColor:colour, fillOpacity:0.2});\n\nreturn shading;\n}\n\n\n//Define your functions\nvar f1 = function(x) {\n  return a*x+b;\n}\n\nvar f2 = function(x) {\n  return a*x*x + c;\n}\n\nvar f3 = function(x) {\n  return (x-a)*(x-b);\n}\n\nvar f4 = function(x) {\n  return 0.5*(x-a)*(x-b)*(x-c);\n}\n\n\n//Plot the graph and do shading\nswitch(q) {\n    case 1:\n        brd.create('functiongraph', [f1]);\n        shade(f1,x1,x2, 'red');\n        break;\n    case 2:\n        brd.create('functiongraph', [f2]);\n        shade(f2,x1,x2,'red');\n        break;\n    case 3:\n        brd.create('functiongraph', [f3]);\n        shade(f3,x1,x2,'red');\n        shade(f3,x2,x2+2,'green');\n        break;\n    case 4:\n        brd.create('functiongraph', [f4]);\n        shade(f4,x1,x2,'red');\n        shade(f4,x2,x2+2,'green');\n        break\n}\n\n\n\nreturn div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>{plotgraph(2,x21,x22,-5,25,a2,0,c2)}</p>\n<p>This is the graph of the function $f(x) = \\simplify{{a2}*x^2+{c2}}$.</p>\n<p>Use integration to calculate the area of the shaded region. Give your answer without any rounding.</p>\n<p>[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "area2", "maxValue": "area2", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>{plotgraph(3,x31,x32,-6,15,a3,b3,0)}</p>\n<p>This curve has equation $y = \\simplify{x^2-{a3+b3}*x + {a3*b3}}$.</p>\n<p>Calculate the total area of the shaded regions. Give your answer without any rounding.&nbsp;&nbsp;(Remember this is a non-calculator question!)</p>\n<p>[[0]]</p>\n<p></p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "area3", "maxValue": "area3", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}]}]}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}]}