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{"name": "Jo-Ann's copy of Definite Integrals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "preamble": {"css": "", "js": ""}, "variables": {"h": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "definition": "random(2..8)"}, "b": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "definition": "random(1..5)"}, "k": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "k", "definition": "random(1..4)"}, "j": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "j", "definition": "random(-4..0)"}, "l": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "l", "definition": "random(2..6)"}, "a": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "definition": "random(-4..0)"}, "d": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "definition": "random(-4..0)"}, "g": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "definition": "random(2..6)"}, "f": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "definition": "random(1..4)"}, "c": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "definition": "random(2..6)"}}, "advice": "<p>To evaluate a definite integral we must first integrate the function (we do not need to include c, the constant of integration) and then substitute in the given limits.</p>\n<p></p>\n<p><strong>(a)</strong></p>\n<p>$\\int_\\var{a}^\\var{b}(1 + \\var{c}x)\\mathrm{dx} =&nbsp;\\left[x + \\var{c/2}x^2\\right]_\\var{a}^\\var{b}= [(\\var{b})+&nbsp;\\var{c/2}(\\var{b})^2]-[(\\var{a}) + \\var{c/2}(\\var{a})^2]=\\simplify{{b}+ {c/2}{b}^2-{a} - {c/2}{a}^2}$</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>$\\int_\\var{d}^\\var{f} (x^2 + \\var{g}x-\\var{h})\\mathrm{dx}=&nbsp;\\left[\\frac{x^3}{3} + \\var{g/2}x^2-\\var{h}x\\right]_\\var{d}^\\var{f}=[\\frac{(\\var{f})^3}{3} + \\var{g/2}(\\var{f})^2-\\var{h}(\\var{f})]-[\\frac{(\\var{d})^3}{3} + \\var{g/2}(\\var{d})^2-\\var{h}(\\var{d})]=\\var{f^3/3 + g/2*f^2-h*f-(d^3/3 + g/2*d^2-h*d)}$</p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>$\\int_\\var{j}^\\var{k}(x^3-\\var{l}x^2)\\mathrm{dx}=\\left[\\frac{x^4}{4} - \\frac{\\var{l}}{3}x^3\\right]_\\var{j}^\\var{k}=[\\frac{(\\var{k})^4}{4} - \\frac{\\var{l}}{3}(\\var{k})^3]-[\\frac{(\\var{j})^4}{4} - \\frac{\\var{l}}{3}(\\var{j})^3]=\\var{(k^4/4 - l/3*k^3)-(j^4/4 - l/3*j^3)}$</p>", "parts": [{"correctAnswerFraction": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": [], "marks": 1, "mustBeReduced": false, "maxValue": "{b}+{c}{b}^2/2-{a}-{a}^2{c}/2", "variableReplacementStrategy": "originalfirst", "allowFractions": true, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "minValue": "{b}+{c}{b}^2/2-{a}-{a}^2{c}/2", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "prompt": "<p>$\\int_\\var{a}^\\var{b}(1 + \\var{c}x)\\mathrm{dx}$</p>", "showFeedbackIcon": true, "type": "numberentry", "scripts": {}}, {"correctAnswerFraction": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": [], "marks": 1, "mustBeReduced": false, "maxValue": "{f}^3/3+{g}{f}^2/2-{h}{f}-{d}^3/3-{g}{d}^2/2+{h}{d}", "variableReplacementStrategy": "originalfirst", "allowFractions": true, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "minValue": "{f}^3/3+{g}{f}^2/2-{h}{f}-{d}^3/3-{g}{d}^2/2+{h}{d}", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "prompt": "<p>$\\int_\\var{d}^\\var{f} (x^2 + \\var{g}x-\\var{h})\\mathrm{dx}$</p>", "showFeedbackIcon": true, "type": "numberentry", "scripts": {}}, {"correctAnswerFraction": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": [], "marks": 1, "mustBeReduced": false, "maxValue": "{k}^4/4-{l}{k}^3/3-{j}^4/4+{l}{j}^3/3", "variableReplacementStrategy": "originalfirst", "allowFractions": true, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "minValue": "{k}^4/4-{l}{k}^3/3-{j}^4/4+{l}{j}^3/3", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "prompt": "<p>$\\int_\\var{j}^\\var{k}(x^3-\\var{l}x^2)\\mathrm{dx}$</p>", "showFeedbackIcon": true, "type": "numberentry", "scripts": {}}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Evaluate the following definite integrals, giving your answer as a fraction as necessary.</p>", "rulesets": {}, "extensions": [], "tags": [], "name": "Jo-Ann's copy of Definite Integrals", "metadata": {"description": "<p>Definite Integrals</p>\n<p>rebel</p>\n<p>rebelmaths</p>\n<p>Rebelmaths</p>\n<p></p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h", "k", "j", "l"], "type": "question", "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}