// Numbas version: exam_results_page_options {"name": "Praneetha's copy sathasivampillai's copy of 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": [{"rulesets": {}, "preamble": {"css": "", "js": ""}, "name": "Praneetha's copy sathasivampillai's copy of Integration: Calculating the area under a curve", "statement": "

This is a non-calculator question.

", "advice": "

See Lecture 12.4, 12.5, 13.1 and 13.2 for background knowledge and examples. Two main pieces of advice:

\n

1) Do a simple estimate to check for big mistakes.

\n

2) Remember that you have to think about whether the area is a `positive area' or `negative area'.  Thinking about mechanics/distances/changes in position is the most coherent way I know of thinking about this.

", "variables": {"a1": {"definition": "1", "templateType": "anything", "name": "a1", "group": "linear graph (not used)", "description": ""}, "x41": {"definition": "a4", "templateType": "anything", "name": "x41", "group": "Ungrouped variables", "description": ""}, "x31": {"definition": "b3-random(2..3)", "templateType": "anything", "name": "x31", "group": "c quadratic. neg region", "description": ""}, "area3": {"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)", "templateType": "anything", "name": "area3", "group": "c quadratic. neg region", "description": ""}, "c4": {"definition": "b4+2", "templateType": "anything", "name": "c4", "group": "Ungrouped variables", "description": ""}, "x12": {"definition": "random(1..4) + x11", "templateType": "anything", "name": "x12", "group": "linear graph (not used)", "description": ""}, "a2": {"definition": "1", "templateType": "anything", "name": "a2", "group": "b) quadratic. no neg region", "description": ""}, "x21": {"definition": "random(-4..-2)", "templateType": "anything", "name": "x21", "group": "b) quadratic. no neg region", "description": ""}, "c2": {"definition": "random(1..4)", "templateType": "anything", "name": "c2", "group": "b) quadratic. no neg region", "description": ""}, "b4": {"definition": "a4+random(2..3)", "templateType": "anything", "name": "b4", "group": "Ungrouped variables", "description": ""}, "x42": {"definition": "b4", "templateType": "anything", "name": "x42", "group": "Ungrouped variables", "description": ""}, "b3": {"definition": "a3+random(3..4)", "templateType": "anything", "name": "b3", "group": "c quadratic. neg region", "description": ""}, "b1": {"definition": "random(1..3)", "templateType": "anything", "name": "b1", "group": "linear graph (not used)", "description": ""}, "a4": {"definition": "random(-3..-2)", "templateType": "anything", "name": "a4", "group": "Ungrouped variables", "description": ""}, "x11": {"definition": "random(1..2)", "templateType": "anything", "name": "x11", "group": "linear graph (not used)", "description": ""}, "a3": {"definition": "random(-3..-1)", "templateType": "anything", "name": "a3", "group": "c quadratic. neg region", "description": ""}, "x33": {"definition": "x32+2", "templateType": "anything", "name": "x33", "group": "c quadratic. neg region", "description": ""}, "x22": {"definition": "random(3..5)+x21", "templateType": "anything", "name": "x22", "group": "b) quadratic. no neg region", "description": ""}, "x32": {"definition": "b3", "templateType": "anything", "name": "x32", "group": "c quadratic. neg region", "description": ""}, "area2": {"definition": "(a2*x22^3/3+c2*x22)-(a2*x21^3/3+c2*x21)", "templateType": "anything", "name": "area2", "group": "b) quadratic. no neg region", "description": ""}}, "extensions": ["jsxgraph"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

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.

"}, "functions": {"plotgraph": {"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;", "type": "html", "parameters": [["q", "number"], ["x1", "number"], ["x2", "number"], ["ymin", "number"], ["ymax", "number"], ["a", "number"], ["b", "number"], ["c", "number"]]}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a4", "b4", "c4", "x41", "x42"], "parts": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "prompt": "

{plotgraph(2,x21,x22,-5,25,a2,0,c2)}

\n

This is the graph of the function $f(x) = \\simplify{{a2}*x^2+{c2}}$.

\n

Use integration to calculate the area of the shaded region. Give your answer without any rounding.

\n

[[0]]

", "gaps": [{"showFeedbackIcon": true, "allowFractions": true, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain", "minValue": "area2", "showCorrectAnswer": true, "maxValue": "area2", "type": "numberentry", "mustBeReducedPC": 0, "correctAnswerFraction": true, "variableReplacements": [], "mustBeReduced": false, "unitTests": [], "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "marks": "3"}], "showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "scripts": {}, "unitTests": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "prompt": "

{plotgraph(3,x31,x32,-6,15,a3,b3,0)}

\n

This curve has equation $y = \\simplify{x^2-{a3+b3}*x + {a3*b3}}$.

\n

(Remember this is a non-calculator question!)

\n

Calculate the total area of the shaded regions. Give your answer without any rounding.

\n

[[0]]

\n

\n

", "gaps": [{"showFeedbackIcon": true, "allowFractions": true, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain", "minValue": "area3", "showCorrectAnswer": true, "maxValue": "area3", "type": "numberentry", "mustBeReducedPC": 0, "correctAnswerFraction": true, "variableReplacements": [], "mustBeReduced": false, "unitTests": [], "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "marks": "5"}], "showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "scripts": {}, "unitTests": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true}], "tags": [], "variable_groups": [{"variables": ["x11", "x12", "a1", "b1"], "name": "linear graph (not used)"}, {"variables": ["x21", "x22", "a2", "c2", "area2"], "name": "b) quadratic. no neg region"}, {"variables": ["a3", "b3", "x31", "x32", "x33", "area3"], "name": "c quadratic. neg region"}], "type": "question", "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "sathasivampillai umakanthan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2528/"}, {"name": "Praneetha Singh", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2552/"}, {"name": "Kamila Yusufu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2554/"}]}]}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "sathasivampillai umakanthan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2528/"}, {"name": "Praneetha Singh", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2552/"}, {"name": "Kamila Yusufu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2554/"}]}