Question Number 159742 by amin96 last updated on 20/Nov/21
$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{perimeter}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{figure}}\:\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{is}} \\ $$$$\boldsymbol{\mathrm{bounded}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{curves}}\:\boldsymbol{\mathrm{y}}^{\mathrm{3}} =\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}=\sqrt{\mathrm{2}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \\ $$
Answered by mr W last updated on 21/Nov/21
![y=x^(2/3) y=(√(2−x^2 )) is a circle with radius (√2) (√(2−x^2 ))=x^(2/3) ⇒x=±1, y=1 both curves intersect at (−1,1), (1,1). length of curve y=x^(2/3) for x=0 to 1: ∫_0 ^1 (√(1+(y′)^2 ))dx =∫_0 ^1 (√(1+((2/3)x^(−(1/3)) )^2 ))dx =∫_0 ^1 (√(1+(4/(9x^(2/3) ))))dx =[x(1+(4/(9x^(2/3) )))^(3/2) ]_0 ^1 =(1+(4/9))^(3/2) =((13(√(13)))/(27)) so the total perimeter is 2×((13(√(13)))/(27))+((2π(√2))/4) =((26(√(13)))/(27))+((π(√2))/2)](https://www.tinkutara.com/question/Q159804.png)
$${y}={x}^{\frac{\mathrm{2}}{\mathrm{3}}} \\ $$$${y}=\sqrt{\mathrm{2}−{x}^{\mathrm{2}} }\:{is}\:{a}\:{circle}\:{with}\:{radius}\:\sqrt{\mathrm{2}} \\ $$$$\sqrt{\mathrm{2}−{x}^{\mathrm{2}} }={x}^{\frac{\mathrm{2}}{\mathrm{3}}} \\ $$$$\Rightarrow{x}=\pm\mathrm{1},\:{y}=\mathrm{1} \\ $$$${both}\:{curves}\:{intersect}\:{at}\:\left(−\mathrm{1},\mathrm{1}\right),\:\left(\mathrm{1},\mathrm{1}\right). \\ $$$${length}\:{of}\:{curve}\:{y}={x}^{\frac{\mathrm{2}}{\mathrm{3}}} \:{for}\:{x}=\mathrm{0}\:{to}\:\mathrm{1}: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\left({y}'\right)^{\mathrm{2}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\left(\frac{\mathrm{2}}{\mathrm{3}}{x}^{−\frac{\mathrm{1}}{\mathrm{3}}} \right)^{\mathrm{2}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\frac{\mathrm{4}}{\mathrm{9}{x}^{\frac{\mathrm{2}}{\mathrm{3}}} }}{dx} \\ $$$$=\left[{x}\left(\mathrm{1}+\frac{\mathrm{4}}{\mathrm{9}{x}^{\frac{\mathrm{2}}{\mathrm{3}}} }\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$=\left(\mathrm{1}+\frac{\mathrm{4}}{\mathrm{9}}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$$=\frac{\mathrm{13}\sqrt{\mathrm{13}}}{\mathrm{27}} \\ $$$${so}\:{the}\:{total}\:{perimeter}\:{is} \\ $$$$\mathrm{2}×\frac{\mathrm{13}\sqrt{\mathrm{13}}}{\mathrm{27}}+\frac{\mathrm{2}\pi\sqrt{\mathrm{2}}}{\mathrm{4}} \\ $$$$=\frac{\mathrm{26}\sqrt{\mathrm{13}}}{\mathrm{27}}+\frac{\pi\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$
Commented by mr W last updated on 21/Nov/21
