prove-that-0-1-1-x-7-1-3-dx-0-1-1-x-3-1-7-dx-

Question Number 76680 by Tony Lin last updated on 29/Dec/19
$${prove}\:{that}: \\$$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} {dx} \\$$
Commented by mr W last updated on 29/Dec/19
$${see}\:{Q}#\mathrm{76232},\:\mathrm{76146} \\$$$$\int_{{a}} ^{{b}} {y}\left({x}\right)\:{dx}=\int_{{c}} ^{{d}} {x}\left({y}\right)\:{dy} \\$$$${y}=\left(\mathrm{1}−{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\$$$$\Rightarrow{y}^{\mathrm{3}} =\mathrm{1}−{x}^{\mathrm{7}} \\$$$$\Rightarrow{x}=\left(\mathrm{1}−{y}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\$$$${y}\left({a}=\mathrm{0}\right)=\mathrm{1}={d} \\$$$${y}\left({b}=\mathrm{1}\right)=\mathrm{0}={c} \\$$$$\int_{\mathrm{0}} ^{\mathrm{1}} {y}\left({x}\right){dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx} \\$$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {x}\left({y}\right){dy}=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{y}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} {dy} \\$$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} {dx} \\$$
Commented by mr W last updated on 29/Dec/19
$${both}\:{integrals}\:{represent}\:{the}\:{same} \\$$$${shaded}\:{area}: \\$$
Commented by mr W last updated on 29/Dec/19
Commented by Tony Lin last updated on 29/Dec/19
$${thanks}\:{sir} \\$$