Answer to Question #195280 in Calculus for fraulyne rayne

Question #195280

Find the common area enclosed by the following pairs of curves.

r^2=6cos2 theta and r=2cos theta


1
Expert's answer
2021-05-20T13:52:11-0400

"\\displaystyle\nr^2 = 6\\cos^2(\\theta),\\,\\, r=2\\cos{\\theta}\\\\\n\n 6\\cos^2(\\theta) = 2\\cos{\\theta}\\\\\n\n2\\cos{\\theta}(3\\cos{\\theta} - 1) = 0\\\\\n\n\\cos{\\theta} = 0,\\,\\ \\theta = \\frac{\\pi}{2}\\\\\n\n\\cos{\\theta} = \\frac{1}{3},\\,\\, \\theta = \\arccos\\left(\\frac{1}{3}\\right)\\\\\n\n\\begin{aligned}\nA_1 &= \\frac{1}{2}\\int_{\\arccos\\left(\\frac{1}{3}\\right)}^{\\frac{\\pi}{2}} 6\\cos^2(\\theta)\\,\\, \\mathrm{d}\\theta\n\\\\&= 3\\int_{\\arccos\\left(\\frac{1}{3}\\right)}^{\\frac{\\pi}{2}} \\cos^2(\\theta)\\,\\, \\mathrm{d}\\theta\n\\end{aligned}\\\\\n\n\\begin{aligned}\nA_2 &= \\frac{1}{2} \\int_{\\arccos\\left(\\frac{1}{3}\\right)}^{\\frac{\\pi}{2}} \\left(2\\cos{\\theta}\\right)^2 \\,\\, \\mathrm{d}\\theta\n\\\\&= 2\\int_{\\frac{\\pi}{2}}^{\\arccos\\left(\\frac{1}{3}\\right)} \\cos^2{\\theta} \\,\\, \\mathrm{d}\\theta\n\\end{aligned}\\\\\n\n\n\\begin{aligned}\nA &= A_1 - A_2 = 3\\int_{\\arccos\\left(\\frac{1}{3}\\right)}^{\\frac{\\pi}{2}} \\cos^2(\\theta)\\,\\, \\mathrm{d}\\theta - 2\\int_{\\arccos\\left(\\frac{1}{3}\\right)}^{\\frac{\\pi}{2}} cos^2{\\theta} \\,\\, \\mathrm{d}\\theta\n\\\\&= \\frac{\\sin(2\\theta) + 2\\theta}{4}\\biggr\\vert_{\\arccos\\left(\\frac{1}{3}\\right)}^{\\frac{\\pi}{2}}\n\\end{aligned}\\\\\n\n\\sin\\left(\\arccos{x}\\right) = \\sqrt{1 - x^2}\\\\\n\\sin\\left(\\arccos\\left(\\frac{1}{3}\\right)\\right) = \\sqrt{1 - \\frac{1}{9}} = \\frac{2\\sqrt{2}}{3}\\\\\n\n\\begin{aligned}\n\\frac{\\sin(2\\theta) + 2\\theta}{4}\\biggr\\vert_{\\arccos\\left(\\frac{1}{3}\\right)}^{\\frac{\\pi}{2}} &= \\frac{-2\\times \\frac{2\\sqrt{2}}{3} \\times \\frac{1}{3} - 2\\arccos(1\/3)}{4} + 2\\frac{\\pi\/2}{4} \n\\\\&= \\frac{-\\frac{4\\sqrt{2}}{9} + 2\\textrm{csc}^{-1}(3)}{4} = \\frac{\\textrm{csc}^{-1}(3)}{2} - \\frac{\\sqrt{2}}{9}\n\\end{aligned}\\\\\n\n\\therefore A = \\frac{\\textrm{csc}^{-1}(3)}{2} - \\frac{\\sqrt{2}}{9}."


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