{"version":9,"randomSeed":"217ffdb83fd9910be7abebaa89a86c9c","graph":{"viewport":{"xmin":-0.13226664266320767,"ymin":-0.09448411898142717,"xmax":1.0977155232209006,"ymax":1.0678248321782386}},"expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"x^{3}+y^{3}=1"},{"type":"expression","id":"2","color":"#2d70b3","latex":"U=\\left[\\left(0,1\\right),\\left(\\frac{1}{3},1\\right),\\left(\\frac{2}{3},\\frac{11}{12}\\right),\\left(\\frac{4}{5},\\frac{4}{5}\\right),\\left(\\frac{11}{12},\\frac{2}{3}\\right),\\left(1,\\frac{1}{3}\\right),\\left(1,0\\right)\\right]","lines":true},{"type":"expression","id":"3","color":"#6042a6","latex":"L=\\left[\\left(0,1\\right),\\left(\\frac{1}{3},\\frac{47}{48}\\right),\\left(\\frac{1}{2},\\frac{19}{20}\\right),\\left(\\frac{2}{3},\\frac{7}{8}\\right),\\left(1,1\\right)\\cdot\\frac{19}{24},\\left(\\frac{7}{8},\\frac{2}{3}\\right),\\left(\\frac{19}{20},\\frac{1}{2}\\right),\\left(\\frac{47}{48},\\frac{1}{3}\\right),\\left(1,0\\right)\\right]","lines":true},{"type":"expression","id":"5","color":"#000000","latex":"D_{3}\\left(a,b\\right)=\\sqrt[3]{\\left|a.x-b.x\\right|^{3}+\\left|a.y-b.y\\right|^{3}}"},{"type":"text","id":"9","text":"bounds on half-pi"},{"type":"expression","id":"6","color":"#c74440","latex":"\\sum_{i=1}^{\\operatorname{length}\\left(L\\right)-1}D_{3}\\left(L\\left[i\\right],L\\left[i+1\\right]\\right)"},{"type":"expression","id":"7","color":"#2d70b3","latex":"\\sum_{i=1}^{\\operatorname{length}\\left(U\\right)-1}D_{3}\\left(U\\left[i\\right],U\\left[i+1\\right]\\right)"},{"type":"expression","id":"12","color":"#000000","latex":"f\\left(x\\right)=\\sqrt[3]{1-x^{3}}"},{"type":"text","id":"17","text":"quarter-pi integral"},{"type":"expression","id":"13","color":"#c74440","latex":"q=\\int_{0}^{\\sqrt[3]{\\frac{1}{2}}}\\sqrt[3]{1+\\left|f'\\left(x\\right)\\right|^{3}}dx"},{"type":"expression","id":"18","color":"#000000","latex":"4q","labelSize":"medium"}]}}