Find the area shaded underneath each of these curves:
", "advice": "", "rulesets": {}, "extensions": ["jsxgraph"], "variables": {"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"}, "a2": {"name": "a2", "group": "b) quadratic. no neg region", "definition": "1", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "linear graph (not used)", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "b3": {"name": "b3", "group": "c quadratic. neg region", "definition": "a3+random(3..4)", "description": "", "templateType": "anything"}, "x32": {"name": "x32", "group": "c quadratic. neg region", "definition": "b3", "description": "", "templateType": "anything"}, "x11": {"name": "x11", "group": "linear graph (not used)", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "c4": {"name": "c4", "group": "Ungrouped variables", "definition": "b4+random(2..3)", "description": "", "templateType": "anything"}, "SecondArea3": {"name": "SecondArea3", "group": "c quadratic. neg region", "definition": "(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"}, "x31": {"name": "x31", "group": "c quadratic. neg region", "definition": "b3-random(2..3)", "description": "", "templateType": "anything"}, "TotalArea4": {"name": "TotalArea4", "group": "Ungrouped variables", "definition": "(FirstArea4)+(SecondArea4)", "description": "", "templateType": "anything"}, "x21": {"name": "x21", "group": "b) quadratic. no neg region", "definition": "random(-4..-2)", "description": "", "templateType": "anything"}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(-3..-2)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "linear graph (not used)", "definition": "1", "description": "", "templateType": "anything"}, "x42": {"name": "x42", "group": "Ungrouped variables", "definition": "b4", "description": "", "templateType": "anything"}, "x41": {"name": "x41", "group": "Ungrouped variables", "definition": "b4-random(2..3)", "description": "", "templateType": "anything"}, "x12": {"name": "x12", "group": "linear graph (not used)", "definition": "random(1..4) + x11", "description": "", "templateType": "anything"}, "x22": {"name": "x22", "group": "b) quadratic. no neg region", "definition": "random(3..5)+x21", "description": "", "templateType": "anything"}, "FirstArea3": {"name": "FirstArea3", "group": "c quadratic. neg region", "definition": "(x33^3/3 - 0.5*(a3+b3)*x33^2+a3*b3*x33)-(x32^3/3 - 0.5*(a3+b3)*x32^2+a3*b3*x32)", "description": "", "templateType": "anything"}, "FirstArea4": {"name": "FirstArea4", "group": "Ungrouped variables", "definition": "(x43/4-({c4}+{b4}+{a4})*x43^3/3+({b4}{c4}+{a4}{c4}+{b4}{a4})*x43^2/2-{b4}{a4}{c4}x43)-(x42/4-({c4}+{b4}+{a4})*x42^3/3+({b4}{c4}+{a4}{c4}+{b4}{a4})*x42^2/2-{b4}{a4}{c4}x42)", "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"}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "a4+random(2..3)", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "b) quadratic. no neg region", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "c quadratic. neg region", "definition": "random(-3..-1)", "description": "", "templateType": "anything"}, "x43": {"name": "x43", "group": "Ungrouped variables", "definition": "c4", "description": "", "templateType": "anything"}, "x33": {"name": "x33", "group": "c quadratic. neg region", "definition": "x32+2", "description": "", "templateType": "anything"}, "SecondArea4": {"name": "SecondArea4", "group": "Ungrouped variables", "definition": "(x42/4-({c4}+{b4}+{a4})*x42^3/3+({b4}{c4}+{a4}{c4}+{b4}{a4})*x42^2/2-{b4}{a4}{c4}x42)-(x41/4-({c4}+{b4}+{a4})*x41^3/3+({b4}{c4}+{a4}{c4}+{b4}{a4})*x41^2/2-{b4}{a4}{c4}x41)", "description": "", "templateType": "anything"}, "d4": {"name": "d4", "group": "Ungrouped variables", "definition": "c4+2", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a4", "b4", "c4", "d4", "x41", "x42", "x43", "FirstArea4", "SecondArea4", "TotalArea4"], "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", "FirstArea3", "SecondArea3"]}], "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 (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, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "{plotgraph(2,x21,x22,-5,25,a2,0,c2)}
\nThis is the graph of the function $f(x) = \\simplify{{a2}*x^2+{c2}}$.
\nUse integration to calculate the area of the shaded region. Give your answer without any rounding.
\n[[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "{plotgraph(3,x31,x32,-6,15,a3,b3,0)}
\nThis curve has equation $y = \\simplify{x^2-({a3}+{b3})x+{a3}{b3}}$
\n\n\nCalculate the total area of the shaded regions. Give your answer without any rounding.
\n[[0]]
\n\n", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "abs(FirstArea3)+Abs(SecondArea3)", "maxValue": "abs(FirstArea3)+Abs(SecondArea3)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Yuri Anissimov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1705/"}, {"name": "Joshua Calcutt", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3267/"}]}]}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Yuri Anissimov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1705/"}, {"name": "Joshua Calcutt", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3267/"}]}