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{"name": "Integration: Definite Integrals of Polynomials - Area Under Graph", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: Definite Integrals of Polynomials - Area Under Graph", "tags": [], "metadata": {"description": "<p>Calculating area under curves of the form $ax^2+bx$ and $ax^4+bx^3+cx^2+dx+e$ in a contextualised problem.</p>", "licence": "None specified"}, "statement": "", "advice": "<p>A cross-section is a cutting made across something, here, the fins of a fish. The curve y models the outline of the cross-section. The area of the cross-section can be found by calculating the definite integral between the two boundaries. Therefore,&nbsp;</p>\n<p>a) The area of the dorsal fin is:</p>\n<p>\\[ \\begin{split} \\simplify[!noLeadingMinus]{defint({-3*a}*x^2+{2*b}*x, x ,0 ,{x_upper})} &amp;\\,= \\left[ \\simplify[!all]{{-3*a}/3 x^3+{2*b}/2 x^2}\\right]_0^\\var{x_upper} \\\\ &amp;\\,= \\left[ \\simplify[all, !noLeadingMinus]{{-a}x^3+{b}x^2}\\right]_0^\\var{x_upper} \\\\ &amp;\\,= \\left[ \\simplify[ !noLeadingMinus]{{-a}*{x_upper}^3+{b}*{x_upper}^2}\\right]-\\left[ \\simplify[ !noLeadingMinus]{{-a}*0^3+{b}*0^2}\\right] \\\\ &amp;\\,=&nbsp; \\left[ \\simplify{{-a*{x_upper}^3}+{b*x_upper^2}}\\right]-\\left[ \\simplify{{-a}*0+{b}*0}\\right] \\\\ &amp;\\,= \\var{ans_a} cm^2\\end{split}\\]&nbsp;</p>\n<p>b)Similarly, the area of the pelvic fin is:</p>\n<p>\\[ \\begin{split} \\simplify[!noLeadingMinus]{defint(-x^4+{4*c}*x^3-{3*d}*x^2+{2*k}*x-{f}, x ,1 ,3.5)} &amp;\\,= \\left[ \\simplify[!all]{-{1/5}* x^5+{4*c}/4*x^4-{3*d}/3*x^3+{2*k}/2*x^2-{f}*x }\\right]_1^{3.5} \\\\ &amp;\\,= \\left[ \\simplify[unitFactor]{-{1/5}* x^5+{4*c/4}*x^4-{3*d/3}*x^3+{2*k/2}*x^2-{f}*x }\\right]_1^{3.5} \\\\ &amp;\\,= \\left[ \\simplify[unitFactor]{-{1/5}* 3.5^5+{4*c/4}*3.5^4-{3*d/3}*3.5^3+{2*k/2}*3.5^2-{f}*3.5 }\\right] - \\left[ \\simplify[unitFactor]{-{1/5}* 1^5+{4*c/4}*1^4-{3*d/3}*1^3+{2*k/2}*1^2-{f}*1 }\\right] \\\\ &amp;\\,= \\left[ \\simplify[unitFactor, fractionNumbers]{-{1*3.5^5/5}+{4*c*3.5^4/4}-{3*d*3.5^3/3}+{2*k*3.5^2/2}-{f*3.5} }\\right] - \\left[ \\simplify[unitFactor, fractionNumbers]{-{1* 1^5/5}+{4*c*1^4/4}-{3*d*1^3/3}+{2*k*1^2/2}-{f*1} }\\right] \\\\ &amp;\\,= \\left[ \\simplify{-{1*3.5^5/5}+{4*c*3.5^4/4}-{3*d*3.5^3/3}+{2*k*3.5^2/2}-{f*3.5} }\\right] - \\left[ \\simplify{-{1* 1^5/5}+{4*c*1^4/4}-{3*d*1^3/3}+{2*k*1^2/2}-{f*1} }\\right] \\\\ &amp;\\,= \\simplify{ {-{1*3.5^5/5}+{4*c*3.5^4/4}-{3*d*3.5^3/3}+{2*k*3.5^2/2}-{f*3.5} } - {-{1* 1^5/5}+{4*c*1^4/4}-{3*d*1^3/3}+{2*k*1^2/2}-{f*1} }} \\\\ &amp;\\,= \\var{ansb} cm^2 \\text{(2 d.p.)} \\end{split}\\]</p>\n<p></p>", "rulesets": {}, "extensions": [], 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"", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Unnamed group", "definition": "random (15..20)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Unnamed group", "definition": "random (10..15)", "description": "", "templateType": "anything", "can_override": false}, "ansb": {"name": "ansb", "group": "Unnamed group", "definition": "precround(0.2*(1-3.5^5)+c*(3.5^4-1)+d*(1-3.5^3)+k*(3.5^2-1)+f*(1-3.5),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "x_upper=ceil(x_upper)", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "x_upper", "ans_a"], "variable_groups": [{"name": "Unnamed group", "variables": ["c", "d", "k", "f", "ansb"]}], "functions": {}, "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": "<p>The cross-section of the dorsal fin of a fish can be modelled by the curve $\\simplify[all,!noLeadingMinus]{y=-3*{a}x^2+2*{b}x}$ for $0\\le\\ x\\le \\var{x_upper}$, where all units are measured in centimetres.</p>\n<p>Calculate the area of the dorsal fin of this fish.</p>\n<p><em>Give your answer rounded to 2 decimal places, if needed.</em></p>\n<p>[[0]] $\\mathrm{cm}^2$</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans_a}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "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": "<p>The cross-section of the pelvic fin of this fish is modelled by the curve&nbsp; \\[ \\simplify[all,!noLeadingMinus]{ y=-x^4+4*{c}x^3-3*{d}x^2+2*{k}x-{f}}\\]&nbsp;for $1\\le\\ x\\le3.5$.</p>\n<p>Calculate the area of the pelvic fin of the fish.</p>\n<p><em>Give your answer rounded to 2 decimal places, if needed.</em></p>\n<p>[[0]] $\\mathrm{cm}^2$</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ansb}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, 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