This is a non-calculator question.
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", "$\\int \\simplify{{a2}*x^2 + {c2}} \\, \\textrm{d}x$
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"], "displayColumns": 0}], "prompt": "(i) {plotgraph(1,x11,x12,-1,10,a1,b1,0)}
\nThis is the graph of the function $f(x) = \\simplify{{a1}*x+{b1}}$.
\nWhich integral corresponds to the area of the shaded region? [[0]]
\n\n(ii) {plotgraph(2,x21,x22,-5,25,a2,0,c2)}
\nThis is the graph of the function $f(x) = \\simplify{{a2}*x^2+{c2}}$.
\nWhich integral will calculate the area of the left region? [[1]]
\nWhich integral gives the total area of both shaded regions? [[2]]
\n\n(iii) {plotgraph(3,x31,x32,-3,7,a3,b3,0)}
\nThis curve has equation $y = \\frac{1}{2}\\simplify{(x-{a3})*(x-{b3})}$.
\nWhich integral gives the area of the left shaded region? [[3]]
\nWhich of these calculates the total area of the two shaded regions? [[4]]
\n\n(iv) {plotgraph(4,x41,x42,-3,7,a4,b4,c4)}
\nThis is the graph of some function $f(x)$.
\nWhich of the following gives the total area of the shaded regions? [[5]]
\n", "customName": ""}], "functions": {"plotgraph": {"parameters": [["q", "number"], ["x1", "number"], ["x2", "number"], ["ymin", "number"], ["ymax", "number"], ["a", "number"], ["b", "number"], ["c", "number"]], "type": "html", "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 '300px',\n '250px',\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 });\n/*var yticks = brd.create('ticks',[yaxis,1],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\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 0.5*(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 shade(f2,x2,x2+2,'green');\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;", "language": "javascript"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "Graphs are given with areas underneath them shaded. The student is asked to select the correct integral which calculates its area.
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