// Numbas version: exam_results_page_options
{"name": "Integration: Definite Integrals 5a - Area between 2 graphs", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: Definite Integrals 5a - Area between 2 graphs", "tags": [], "metadata": {"description": "<p>Calculating the area enclosed between a linear function and a quadratic function by integration. The limits (points of intersection) are given in the question.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the area enclosed by $\\simplify{y={m}x+{c_1}}$ and $\\simplify{y=x^2+{b}x+{c_2}}$ between $x=\\var{cp1}$ and $x=\\var{cp2}$.</p>", "advice": "<p>When finding the area between two&nbsp;functions it can be helpful to sketch the graph of these functions to have a better understanding of the area you are trying to find. In this case, we have $\\simplify{y={m}x+{c_1}}$ and $\\simplify{y=x^2+{b}x+{c_2}}$:</p>\n<p>{geogebra_applet('https://www.geogebra.org/m/cxemr4pd',defs)}</p>\n<p>We can see that&nbsp;the area we are being asked to find is bounded above by the line&nbsp;$\\simplify{y={m}x+{c_1}}$, and bounded below by the curve&nbsp;$\\simplify{y=x^2+{b}x+{c_2}}$. Additionally, the $x$-values we are told to find the area between are the points of intersection between these two functions.</p>\n<p>Hence, by finding the difference between the integrals</p>\n<p>\\[\\simplify{defint({m}x+{c_1},x,{cp1},{cp2})}\\quad \\text{and} \\quad \\simplify{defint(x^2+{b}x+{c_2},x,{cp1},{cp2})} ,\\]</p>\n<p>this will give us the area we are being asked to find.</p>\n<p>So,</p>\n<p>\\[ \\begin{split}\\simplify{defint({m}x+{c_1},x,{cp1},{cp2})}-\\simplify{defint(x^2+{b}x+{c_2},x,{cp1},{cp2})} &amp;\\,= \\simplify{defint(({m}x+{c_1})-(x^2+{b}x+{c_2}),x,{cp1},{cp2})},\\\\ &amp;\\,=\\simplify[all,!noLeadingMinus]{defint(-x^2+{m-b}x+{c_1-c_2},x,{cp1},{cp2})}. \\end{split} \\]</p>\n<p>Hence,</p>\n<p>{advice}</p>\n<p></p>", "rulesets": {}, "extensions": ["geogebra"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-6..6)", "description": "", "templateType": "anything", "can_override": false}, "c_1": {"name": "c_1", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "cp1": {"name": "cp1", "group": "Ungrouped variables", "definition": "(-(b-m)-sqrt(dis))/2", "description": "", "templateType": "anything", "can_override": false}, "cp2": {"name": "cp2", "group": "Ungrouped variables", "definition": "(-(b-m)+sqrt(dis))/2", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "c_2": {"name": "c_2", "group": "Ungrouped variables", "definition": "random(-8..8 except [0,c_1])", "description": "", "templateType": "anything", "can_override": false}, "dis": {"name": "dis", "group": "Ungrouped variables", "definition": "(b-m)^2-4*(c_2-c_1)", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "-(cp2^3/3)+(m-b)cp2^2/2+(c_1-c_2)*cp2+(cp1)^3/3-(m-b)cp1^2/2-(c_1-c_2)*cp1", "description": "", "templateType": "anything", "can_override": false}, "sol2dp": {"name": "sol2dp", "group": "Ungrouped variables", "definition": "precround(sol,2)", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(sol=round(sol),'{simplified}',if(sol=sol2dp,'{simplifyy}','{simplify}'))", "description": "", "templateType": "anything", "can_override": false}, "simplified": {"name": "simplified", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split}\\\\simplify[all, !noLeadingMinus]{defint(-x^2+{m-b}x+{c_1-c_2},x,{cp1},{cp2})} &amp;\\\\,= \\\\left[\\\\simplify[all, !noLeadingMinus]{-x^3/3+{m-b}x^2/2+{c_1-c_2}x} \\\\right]_\\\\var{cp1}^\\\\var{cp2},\\\\\\\\ &amp;\\\\,=\\\\left[\\\\simplify[all, !collectNumbers,fractionNumbers,!noLeadingMinus]{-{cp2^3/3}+{(m-b)*cp2^2/2}+{(c_1-c_2)*cp2}}\\\\right]-\\\\left[\\\\simplify[all, !collectNumbers,fractionNumbers,!noLeadingMinus]{-{cp1^3/3}+{(m-b)*cp1^2/2}+{(c_1-c_2)*cp1}}\\\\right]\\\\\\\\ &amp;\\\\,= \\\\left(\\\\simplify[all, fractionNumbers]{{-(cp2^3/3)+(m-b)cp2^2/2+(c_1-c_2)*cp2}}\\\\right)-\\\\left(\\\\simplify[all, fractionNumbers]{{-(cp1)^3/3+(m-b)cp1^2/2+(c_1-c_2)*cp1}}\\\\right)\\\\\\\\ &amp;\\\\,=\\\\simplify[all, fractionNumbers]{{-(cp2^3/3)+(m-b)cp2^2/2+(c_1-c_2)*cp2+(cp1)^3/3-(m-b)cp1^2/2-(c_1-c_2)*cp1}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "simplify": {"name": "simplify", "group": "Ungrouped variables", "definition": "safe(\"<p>\\\\[ \\\\begin{split}\\\\simplify[all, !noLeadingMinus]{defint(-x^2+{m-b}x+{c_1-c_2},x,{cp1},{cp2})} &amp;\\\\,= \\\\left[\\\\simplify[all, !noLeadingMinus]{-x^3/3+{m-b}x^2/2+{c_1-c_2}x} \\\\right]_\\\\var{cp1}^\\\\var{cp2},\\\\\\\\ &amp;\\\\,=\\\\left[\\\\simplify[all, !collectNumbers,fractionNumbers,!noLeadingMinus]{-{cp2^3/3}+{(m-b)*cp2^2/2}+{(c_1-c_2)*cp2}}\\\\right]-\\\\left[\\\\simplify[all, !collectNumbers,fractionNumbers,!noLeadingMinus]{-{cp1^3/3}+{(m-b)*cp1^2/2}+{(c_1-c_2)*cp1}}\\\\right]\\\\\\\\ &amp;\\\\,= \\\\left(\\\\simplify[all, fractionNumbers]{{-(cp2^3/3)+(m-b)cp2^2/2+(c_1-c_2)*cp2}}\\\\right)-\\\\left(\\\\simplify[all, fractionNumbers]{{-(cp1)^3/3+(m-b)cp1^2/2+(c_1-c_2)*cp1}}\\\\right)\\\\\\\\ &amp;\\\\,=\\\\simplify[all, fractionNumbers]{{-(cp2^3/3)+(m-b)cp2^2/2+(c_1-c_2)*cp2+(cp1)^3/3-(m-b)cp1^2/2-(c_1-c_2)*cp1}}\\\\\\\\ &amp;\\\\,=\\\\var{sol2dp} \\\\,\\\\text{(2 d.p.)}. \\\\end{split} \\\\]</p>\")", "description": "", "templateType": "long string", "can_override": false}, "simplifyy": {"name": "simplifyy", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split}\\\\simplify[all,!noLeadingMinus]{defint(-x^2+{m-b}x+{c_1-c_2},x,{cp1},{cp2})} &amp;\\\\,= \\\\left[\\\\simplify[all,!noLeadingMinus]{-x^3/3+{m-b}x^2/2+{c_1-c_2}x} \\\\right]_\\\\var{cp1}^\\\\var{cp2},\\\\\\\\ &amp;\\\\,=\\\\left[\\\\simplify[all, !collectNumbers,fractionNumbers,!noLeadingMinus]{-{cp2^3/3}+{(m-b)*cp2^2/2}+{(c_1-c_2)*cp2}}\\\\right]-\\\\left[\\\\simplify[all, !collectNumbers,fractionNumbers,!noLeadingMinus]{-{cp1^3/3}+{(m-b)*cp1^2/2}+{(c_1-c_2)*cp1}}\\\\right]\\\\\\\\ &amp;\\\\,= \\\\left(\\\\simplify[all, fractionNumbers]{{-(cp2^3/3)+(m-b)cp2^2/2+(c_1-c_2)*cp2}}\\\\right)-\\\\left(\\\\simplify[all, fractionNumbers]{{-(cp1)^3/3+(m-b)cp1^2/2+(c_1-c_2)*cp1}}\\\\right)\\\\\\\\ &amp;\\\\,=\\\\simplify[all, fractionNumbers]{{-(cp2^3/3)+(m-b)cp2^2/2+(c_1-c_2)*cp2+(cp1)^3/3-(m-b)cp1^2/2-(c_1-c_2)*cp1}}\\\\\\\\ &amp;\\\\,=\\\\var{sol2dp} \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "defs": {"name": 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