Calculating the area enclosed between a linear function and a quadratic function by integration. The limits (points of intersection) are not given in the question and must be calculated.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Find the area enclosed by $\\simplify{y={m}x+{c_1}}$ and $\\simplify{y=x^2+{b}x+{c_2}}$ using a double integral.
\n\n{geogebra_applet('https://www.geogebra.org/m/fyxr2fqj',defs)}
", "advice": "a)
\nTo find the points of intersection, we will solve the given eqautions simultaneously. We equate the two and solve for $x$ first:
\n\\[ \\begin{split} \\simplify{x^2+{b}x+{c_2}} &\\,= \\simplify{{m}x+{c_1}}, \\\\ \\simplify{x^2+{b-m}x+{c_2-c_1}} &\\,=0, \\\\ \\simplify{(x-{cp1})(x-{cp2})}&\\,=0. \\end{split} \\]
\nTherefore, the points of intersection are when $\\simplify{x1={cp1}}$ and $\\simplify{x2={cp2}}$.
\nNow by pluging-in these values into $\\simplify{y={m}x+{c_1}}$ we get $\\simplify{y_1 = {y1}}$ and $\\simplify{y_2 = {y2}}$.
\n\nb)
\nWe are asked to integrate $y$ first. So, $x$ can vary freely in the range we found above, and $y$ will vary depending on $x$. So the relevant boundaries of the variables are
\n\\[ \\var{x1} < x < \\var{x2} \\quad \\mbox{ and }\\quad \\simplify{x^2+{b}x+{c_2}} < y < \\simplify{{m}x+{c_1}}\\simplify{x^2+{b}x+{c_2}}.\\]
\nThen the integral
\n\\[\\int_{\\var{x1}}^{\\var{x2}} \\int_{\\simplify{x^2+{b}x+{c_2}}}^{\\simplify{{m}x+{c_1}}}\\, \\mathrm{d}y\\, \\mathrm{d}x\\]
\ncan be used to compute the shaded area.
\n\nc)
\nWe start with the inner integral first
\n\\[\\int_{\\simplify{x^2+{b}x+{c_2}}}^{\\simplify{{m}x+{c_1}}}\\, \\mathrm{d}y = y|_{\\simplify{x^2+{b}x+{c_2}}}^{\\simplify{{m}x+{c_1}}} = \\simplify{{m}x+{c_1}} - (\\simplify{x^2+{b}x+{c_2}}) = \\simplify{-x^2 + {m-b}x + {c_1-c_2}}.\\]
\n\nThen we get
\n\\[\\int_{\\var{x1}}^{\\var{x2}} \\int_{\\simplify{x^2+{b}x+{c_2}}}^{\\simplify{{m}x+{c_1}}}\\, \\mathrm{d}y\\, \\mathrm{d}x =\\] {advice}
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\\\\[ \\\\begin{split}\\\\simplify[all, !noLeadingMinus]{defint(-x^2+{m-b}x+{c_1-c_2},x,{cp1},{cp2})} &\\\\,= \\\\left[\\\\simplify[all, !noLeadingMinus]{-x^3/3+{m-b}x^2/2+{c_1-c_2}x} \\\\right]_\\\\var{cp1}^\\\\var{cp2},\\\\\\\\ &\\\\,=\\\\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]\\\\\\\\ &\\\\,= \\\\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)\\\\\\\\ &\\\\,=\\\\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} \\\\]
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\n$x_1=$[[0]], $y_1=$[[1]]
\n$x_2=$[[2]], $y_2=$ [[3]]
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\n[[0]] (Give your answers to 2 decimal places where necessary)
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