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.
"}, "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"]}], "name": "Integration: Calculating the area under a curve", "parts": [{"type": "gapfill", "marks": 0, "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", "marks": "2", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "mustBeReducedPC": 0, "mustBeReduced": false, "scripts": {}, "customName": "", "variableReplacements": [], "showFractionHint": true, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "maxValue": "area2", "minValue": "area2", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "correctAnswerFraction": true}], "scripts": {}, "customName": "", "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "sortAnswers": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true}, {"type": "gapfill", "marks": 0, "prompt": "{plotgraph(3,x31,x32,-6,15,a3,b3,0)}
\nThis curve has equation $y = \\simplify{x^2-{a3+b3}*x + {a3*b3}}$.
\n(Remember this is a non-calculator question!)
\nCalculate the total area of the shaded regions. Give your answer without any rounding.
\n[[0]]
\n\n", "gaps": [{"type": "numberentry", "marks": "4", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "mustBeReducedPC": 0, "mustBeReduced": false, "scripts": {}, "customName": "", "variableReplacements": [], "showFractionHint": true, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "maxValue": "area3", "minValue": "area3", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "correctAnswerFraction": true}], "scripts": {}, "customName": "", "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "sortAnswers": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true}], "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "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;"}}, "variables": {"a2": {"name": "a2", "templateType": "anything", "description": "", "definition": "1", "group": "b) quadratic. no neg region"}, "a1": {"name": "a1", "templateType": "anything", "description": "", "definition": "1", "group": "linear graph (not used)"}, "x22": {"name": "x22", "templateType": "anything", "description": "", "definition": "random(3..5)+x21", "group": "b) quadratic. no neg region"}, "b1": {"name": "b1", "templateType": "anything", "description": "", "definition": "random(1..3)", "group": "linear graph (not used)"}, "a3": {"name": "a3", "templateType": "anything", "description": "", "definition": "random(-3..-1)", "group": "c quadratic. neg region"}, "b3": {"name": "b3", "templateType": "anything", "description": "", "definition": "a3+random(3..4)", "group": "c quadratic. neg region"}, "x33": {"name": "x33", "templateType": "anything", "description": "", "definition": "x32+2", "group": "c quadratic. neg region"}, "x31": {"name": "x31", "templateType": "anything", "description": "", "definition": "b3-random(2..3)", "group": "c quadratic. neg region"}, "a4": {"name": "a4", "templateType": "anything", "description": "", "definition": "random(-3..-2)", "group": "Ungrouped variables"}, "b4": {"name": "b4", "templateType": "anything", "description": "", "definition": "a4+random(2..3)", "group": "Ungrouped variables"}, "area3": {"name": "area3", "templateType": "anything", "description": "", "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)", "group": "c quadratic. neg region"}, "x12": {"name": "x12", "templateType": "anything", "description": "", "definition": "random(1..4) + x11", "group": "linear graph (not used)"}, "x42": {"name": "x42", "templateType": "anything", "description": "", "definition": "b4", "group": "Ungrouped variables"}, "c4": {"name": "c4", "templateType": "anything", "description": "", "definition": "b4+2", "group": "Ungrouped variables"}, "x32": {"name": "x32", "templateType": "anything", "description": "", "definition": "b3", "group": "c quadratic. neg region"}, "x11": {"name": "x11", "templateType": "anything", "description": "", "definition": "random(1..2)", "group": "linear graph (not used)"}, "x41": {"name": "x41", "templateType": "anything", "description": "", "definition": "a4", "group": "Ungrouped variables"}, "x21": {"name": "x21", "templateType": "anything", "description": "", "definition": "random(-4..-2)", "group": "b) quadratic. no neg region"}, "c2": {"name": "c2", "templateType": "anything", "description": "", "definition": "random(1..4)", "group": "b) quadratic. no neg region"}, "area2": {"name": "area2", "templateType": "anything", "description": "", "definition": "(a2*x22^3/3+c2*x22)-(a2*x21^3/3+c2*x21)", "group": "b) quadratic. no neg region"}}, "statement": "This is a non-calculator question.
", "ungrouped_variables": ["a4", "b4", "c4", "x41", "x42"], "advice": "See Lecture 13.3 and 13.5 for background knowledge and examples. (You may need 13.1 and 12.3 for more background knowledge).
\nMain advice is to go through the steps given in lectures: make an estimate, establish the integral(s) required, do the calculations, check your answer.
", "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}