Evaluate the following double integrals.
\nInput your answer as an integer or a fraction, not as a decimal.
", "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "metadata": {"description": "Two double integrals with numerical limits
", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"variableReplacements": [], "gaps": [{"notallowed": {"strings": ["."], "message": "Input all numbers in your answer as integers or fractions, not as decimals.
", "showStrings": false, "partialCredit": 0}, "showFeedbackIcon": true, "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showpreview": true, "variableReplacements": [], "answersimplification": "std", "checkingtype": "absdiff", "answer": "{c*b*(a-1)+(4*d*b*b/4)*(a*a-1)}", "marks": 4, "type": "jme", "vsetrange": [0, 1], "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "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]]
", "variableReplacementStrategy": "originalfirst"}, {"variableReplacements": [], "gaps": [{"answersimplification": "fractionnumbers", "showFeedbackIcon": true, "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showpreview": true, "variableReplacements": [], "checkingtype": "absdiff", "answer": "{-h^(f+1)*((-1)^g-1)/(g*(f+1))}", "marks": 4, "type": "jme", "vsetrange": [0, 1], "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "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]]
", "variableReplacementStrategy": "originalfirst"}], "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "preamble": {"js": "", "css": ""}, "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}