// Numbas version: exam_results_page_options {"name": "Luis's copy of Evaluate double integrals with numerical limits", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Luis's copy of Evaluate double integrals with numerical limits", "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"showFeedbackIcon": true, "prompt": "
\\[I = \\int^\\var{a}_{y=1} \\int^\\var{b}_{x=0} \\left(\\var{c}+\\simplify[std]{{4*d}*x*y} \\right) \\; \\mathrm{d}x \\, \\mathrm{d}y \\]
\n$I =$ [[0]]
", "gaps": [{"showFeedbackIcon": true, "checkvariablenames": false, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "answersimplification": "std", "variableReplacements": [], "type": "jme", "expectedvariablenames": [], "scripts": {}, "marks": 4, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "answer": "{c*b*(a-1)+(4*d*b*b/4)*(a*a-1)}", "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "Input all numbers in your answer as integers or fractions, not as decimals.
", "strings": ["."], "showStrings": false, "partialCredit": 0}}], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "marks": 0}, {"showFeedbackIcon": true, "prompt": "\\[ I = \\int^\\pi_{x=0} \\int^\\var{h}_{y=0} \\simplify[std]{y^{f}sin({g}x)} \\; \\mathrm{d}y \\, \\mathrm{d}x \\]
\n$I=$ [[0]]
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"}, "tags": [], "advice": "We proceed to evaluate the double-integral:
\n\\begin{align}
I &= \\int^\\var{a}_1 \\int^\\var{b}_0 \\left(\\var{c}+\\simplify[std]{{4*d}*x*y} \\right) \\; \\mathrm{d}x \\, \\mathrm{d}y \\\\
&= \\int^\\var{a}_1 \\left[\\simplify[std]{{c}x+{2*d}*y*x^2} \\right]_{x=0}^\\var{b} \\; \\mathrm{d}y \\\\
&= \\int^\\var{a}_1 \\left(\\simplify[std]{{c*b}+{2*d*b^2}*y} \\right) \\; \\mathrm{d}y \\\\
&= \\left[\\simplify[std]{{c*b}y+{d*b^2}*y^2} \\right]^\\var{a}_1 \\; \\mathrm{d}y \\\\
&= \\simplify[std]{{c*b*a}+{d*b^2*a^2}-{c*b}-{d*b^2}} \\\\
&= \\simplify[std]{{(c*b*a)+(d*b^2*a^2)-(c*b)-(d*b^2)}}
\\end{align}
\\begin{align}
I &= \\int^\\pi_0 \\int^\\var{h}_0 \\simplify[std]{y^{f}sin({g}x)} \\; \\mathrm{d}y \\, \\mathrm{d}x \\\\
&= \\int^\\pi_0 \\left[\\simplify[std]{(1/{f+1})*y^{f+1}*sin({g}x)}\\right]_{y=0}^\\var{h} \\; \\mathrm{d}x \\\\[0.5em]
&= \\int^\\pi_0 \\simplify[std]{({h}^{f+1}/{f+1})*sin({g}x)} \\; \\mathrm{d}x \\\\[0.5em]
&= \\simplify[std]{({h}^{f+1}/{f+1})}\\left[\\simplify[std]{-1/{g}*cos({g}x)}\\right]^\\pi_0 \\\\[0.5em]
&= -\\simplify[std]{({h}^{f+1}/{g*(f+1)})} \\left(\\simplify[std]{{(-1)^g}}-1 \\right) \\\\[0.5em]
&= \\simplify[fractionnumbers]{{-{h}^({f+1})*((-1)^{g}-1)/({g*(f+1)})}}
\\end{align}
Evaluate the following double integrals.
\nInput your answer as an integer or a fraction, not as a decimal.
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