Calculating the area enclosed between a cosine function and a quadratic function by integration. The limits (points of intersection) are given in the question.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Find the area enclosed by $\\simplify{y=cos(pi*x/{a})}$ and $\\simplify{y=x^2-{a^2+1}}$, between $x=\\var{-a}$ and $x=\\var{a}$.
", "advice": "When finding the area between two 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 $\\simplify{y=cos(pi x/{a})}$ and $\\simplify{y=x^2-{a^2+1}}$:
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\nWe can see that the lower bound of the area is the curve $\\simplify{y=x^2-{a^2+1}}$ and the upper bound is the curve $\\simplify{y=cos(pi x/{a})}$, and the $x$-values we are evaluating between are where the curves intersect.
\nTherefore, by finding the difference between the integrals
\n\\[\\simplify{defint(cos(pi x/{a}),x,{-a},{a})}\\quad \\text{and} \\quad \\simplify{defint(x^2-{a^2+1},x,{-a},{a})} ,\\]
\nthis will give us the area we are being asked to find.
\nSo,
\n\\[ \\begin{split}\\simplify{defint(cos(pi x/{a}),x,{-a},{a})}-\\simplify{defint(x^2-{a^2+1},x,{-a},{a})} &\\,= \\simplify{defint((cos(pi x/{a}))-(x^2-{a^2+1}),x,{-a},{a})},\\\\ &\\,=\\simplify{defint(cos(pi x/{a})-x^2+{a^2+1},x,{-a},{a})}. \\end{split} \\]
\nHence,
\n\n{advice}
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\\n\"", "description": "", "templateType": "long string", "can_override": false}, "simplify": {"name": "simplify", "group": "Ungrouped variables", "definition": "\"\\\\[ \\\\begin{split} \\\\simplify{defint(cos(pi x/{a})-x^2+{a^2+1},x,{-a},{a})} &\\\\,= \\\\left[\\\\simplify{{a}/pi sin(pi x/{a})-x^3/3+{a^2+1}x}\\\\right]_\\\\var{-a}^\\\\var{a}\\\\\\\\ &\\\\,= \\\\left[\\\\simplify[all,fractionNumbers,!zeroTerm,!collectNumbers]{0-{a^3/3}+{(a^2+1)(a)}}\\\\right]-\\\\left[\\\\simplify[all,fractionNumbers,!zeroTerm,!collectNumbers]{0-{(-a)^3/3}+{(a^2+1)(-a)}}\\\\right] \\\\\\\\ &\\\\,=\\\\simplify[all,fractionNumbers]{{2((a^2+1)(a)-(a)^3/3)}}\\\\\\\\&\\\\,=\\\\var{soldp}\\\\,\\\\text{(2dp)}\\\\end{split} \\\\]
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